The following may well be the most controversial dilemma in the history of decision theory:

    A superintelligence from another galaxy, whom we shall call Omega, comes to Earth and sets about playing a strange little game.  In this game, Omega selects a human being, sets down two boxes in front of them, and flies away.

    Box A is transparent and contains a thousand dollars.
    Box B is opaque, and contains either a million dollars, or nothing.

    You can take both boxes, or take only box B.

    And the twist is that Omega has put a million dollars in box B iff Omega has predicted that you will take only box B.

    Omega has been correct on each of 100 observed occasions so far - everyone who took both boxes has found box B empty and received only a thousand dollars; everyone who took only box B has found B containing a million dollars.  (We assume that box A vanishes in a puff of smoke if you take only box B; no one else can take box A afterward.)

    Before you make your choice, Omega has flown off and moved on to its next game.  Box B is already empty or already full.

    Omega drops two boxes on the ground in front of you and flies off.

    Do you take both boxes, or only box B?

    And the standard philosophical conversation runs thusly:

    One-boxer:  "I take only box B, of course.  I'd rather have a million than a thousand."

    Two-boxer:  "Omega has already left.  Either box B is already full or already empty.  If box B is already empty, then taking both boxes nets me $1000, taking only box B nets me $0.  If box B is already full, then taking both boxes nets $1,001,000, taking only box B nets $1,000,000.  In either case I do better by taking both boxes, and worse by leaving a thousand dollars on the table - so I will be rational, and take both boxes."

    One-boxer:  "If you're so rational, why ain'cha rich?"

    Two-boxer:  "It's not my fault Omega chooses to reward only people with irrational dispositions, but it's already too late for me to do anything about that."

    There is a large literature on the topic of Newcomblike problems - especially if you consider the Prisoner's Dilemma as a special case, which it is generally held to be.  "Paradoxes of Rationality and Cooperation" is an edited volume that includes Newcomb's original essay.  For those who read only online material, this PhD thesis summarizes the major standard positions.

    I'm not going to go into the whole literature, but the dominant consensus in modern decision theory is that one should two-box, and Omega is just rewarding agents with irrational dispositions.  This dominant view goes by the name of "causal decision theory".

    As you know, the primary reason I'm blogging is that I am an incredibly slow writer when I try to work in any other format.  So I'm not going to try to present my own analysis here.  Way too long a story, even by my standards.

    But it is agreed even among causal decision theorists that if you have the power to precommit yourself to take one box, in Newcomb's Problem, then you should do so.  If you can precommit yourself before Omega examines you; then you are directly causing box B to be filled.

    Now in my field - which, in case you have forgotten, is self-modifying AI - this works out to saying that if you build an AI that two-boxes on Newcomb's Problem, it will self-modify to one-box on Newcomb's Problem, if the AI considers in advance that it might face such a situation.  Agents with free access to their own source code have access to a cheap method of precommitment.

    What if you expect that you might, in general, face a Newcomblike problem, without knowing the exact form of the problem?  Then you would have to modify yourself into a sort of agent whose disposition was such that it would generally receive high rewards on Newcomblike problems.

    But what does an agent with a disposition generally-well-suited to Newcomblike problems look like?  Can this be formally specified?

    Yes, but when I tried to write it up, I realized that I was starting to write a small book.  And it wasn't the most important book I had to write, so I shelved it.  My slow writing speed really is the bane of my existence.  The theory I worked out seems, to me, to have many nice properties besides being well-suited to Newcomblike problems.  It would make a nice PhD thesis, if I could get someone to accept it as my PhD thesis.  But that's pretty much what it would take to make me unshelve the project.  Otherwise I can't justify the time expenditure, not at the speed I currently write books.

    I say all this, because there's a common attitude that "Verbal arguments for one-boxing are easy to come by, what's hard is developing a good decision theory that one-boxes" - coherent math which one-boxes on Newcomb's Problem without producing absurd results elsewhere.  So I do understand that, and I did set out to develop such a theory, but my writing speed on big papers is so slow that I can't publish it.  Believe it or not, it's true.

    Nonetheless, I would like to present some of my motivations on Newcomb's Problem - the reasons I felt impelled to seek a new theory - because they illustrate my source-attitudes toward rationality.  Even if I can't present the theory that these motivations motivate...

    First, foremost, fundamentally, above all else:

    Rational agents should WIN.

    Don't mistake me, and think that I'm talking about the Hollywood Rationality stereotype that rationalists should be selfish or shortsighted.  If your utility function has a term in it for others, then win their happiness.  If your utility function has a term in it for a million years hence, then win the eon.

    But at any rate, WIN.  Don't lose reasonably, WIN.

    Now there are defenders of causal decision theory who argue that the two-boxers are doing their best to win, and cannot help it if they have been cursed by a Predictor who favors irrationalists.  I will talk about this defense in a moment.  But first, I want to draw a distinction between causal decision theorists who believe that two-boxers are genuinely doing their best to win; versus someone who thinks that two-boxing is the reasonable or the rational thing to do, but that the reasonable move just happens to predictably lose, in this case.  There are a lot of people out there who think that rationality predictably loses on various problems - that, too, is part of the Hollywood Rationality stereotype, that Kirk is predictably superior to Spock.

    Next, let's turn to the charge that Omega favors irrationalists.  I can conceive of a superbeing who rewards only people born with a particular gene, regardless of their choices.  I can conceive of a superbeing who rewards people whose brains inscribe the particular algorithm of "Describe your options in English and choose the last option when ordered alphabetically," but who does not reward anyone who chooses the same option for a different reason.  But Omega rewards people who choose to take only box B, regardless of which algorithm they use to arrive at this decision, and this is why I don't buy the charge that Omega is rewarding the irrational.  Omega doesn't care whether or not you follow some particular ritual of cognition; Omega only cares about your predicted decision.

    We can choose whatever reasoning algorithm we like, and will be rewarded or punished only according to that algorithm's choices, with no other dependency - Omega just cares where we go, not how we got there.

    It is precisely the notion that Nature does not care about our algorithm, which frees us up to pursue the winning Way - without attachment to any particular ritual of cognition, apart from our belief that it wins.  Every rule is up for grabs, except the rule of winning.

    As Miyamoto Musashi said - it's really worth repeating:

    "You can win with a long weapon, and yet you can also win with a short weapon.  In short, the Way of the Ichi school is the spirit of winning, whatever the weapon and whatever its size."

    (Another example:  It was argued by McGee that we must adopt bounded utility functions or be subject to "Dutch books" over infinite times.  But:  The utility function is not up for grabs.  I love life without limit or upper bound:  There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever.  This is a sufficient condition to imply that my utility function is unbounded.  So I just have to figure out how to optimize for that morality.  You can't tell me, first, that above all I must conform to a particular ritual of cognition, and then that, if I conform to that ritual, I must change my morality to avoid being Dutch-booked.  Toss out the losing ritual; don't change the definition of winning.  That's like deciding to prefer $1000 to $1,000,000 so that Newcomb's Problem doesn't make your preferred ritual of cognition look bad.)

    "But," says the causal decision theorist, "to take only one box, you must somehow believe that your choice can affect whether box B is empty or full - and that's unreasonable!  Omega has already left!  It's physically impossible!"

    Unreasonable?  I am a rationalist: what do I care about being unreasonable?  I don't have to conform to a particular ritual of cognition.  I don't have to take only box B because I believe my choice affects the box, even though Omega has already left.  I can just... take only box B.

    I do have a proposed alternative ritual of cognition which computes this decision, which this margin is too small to contain; but I shouldn't need to show this to you.  The point is not to have an elegant theory of winning - the point is to win; elegance is a side effect.

    Or to look at it another way:  Rather than starting with a concept of what is the reasonable decision, and then asking whether "reasonable" agents leave with a lot of money, start by looking at the agents who leave with a lot of money, develop a theory of which agents tend to leave with the most money, and from this theory, try to figure out what is "reasonable".  "Reasonable" may just refer to decisions in conformance with our current ritual of cognition - what else would determine whether something seems "reasonable" or not?

    From James Joyce (no relation), Foundations of Causal Decision Theory:

    Rachel has a perfectly good answer to the "Why ain't you rich?" question.  "I am not rich," she will say, "because I am not the kind of person the psychologist thinks will refuse the money.  I'm just not like you, Irene.  Given that I know that I am the type who takes the money, and given that the psychologist knows that I am this type, it was reasonable of me to think that the $1,000,000 was not in my account.  The $1,000 was the most I was going to get no matter what I did.  So the only reasonable thing for me to do was to take it."

    Irene may want to press the point here by asking, "But don't you wish you were like me, Rachel?  Don't you wish that you were the refusing type?"  There is a tendency to think that Rachel, a committed causal decision theorist, must answer this question in the negative, which seems obviously wrong (given that being like Irene would have made her rich).  This is not the case.  Rachel can and should admit that she does wish she were more like Irene.  "It would have been better for me," she might concede, "had I been the refusing type."  At this point Irene will exclaim, "You've admitted it!  It wasn't so smart to take the money after all."  Unfortunately for Irene, her conclusion does not follow from Rachel's premise.  Rachel will patiently explain that wishing to be a refuser in a Newcomb problem is not inconsistent with thinking that one should take the $1,000 whatever type one is.  When Rachel wishes she was Irene's type she is wishing for Irene's options, not sanctioning her choice.

    It is, I would say, a general principle of rationality - indeed, part of how I define rationality - that you never end up envying someone else's mere choices.  You might envy someone their genes, if Omega rewards genes, or if the genes give you a generally happier disposition.  But Rachel, above, envies Irene her choice, and only her choice, irrespective of what algorithm Irene used to make it.  Rachel wishes just that she had a disposition to choose differently.

    You shouldn't claim to be more rational than someone and simultaneously envy them their choice - only their choice.  Just do the act you envy.

    I keep trying to say that rationality is the winning-Way, but causal decision theorists insist that taking both boxes is what really wins, because you can't possibly do better by leaving $1000 on the table... even though the single-boxers leave the experiment with more money.  Be careful of this sort of argument, any time you find yourself defining the "winner" as someone other than the agent who is currently smiling from on top of a giant heap of utility.

    Yes, there are various thought experiments in which some agents start out with an advantage - but if the task is to, say, decide whether to jump off a cliff, you want to be careful not to define cliff-refraining agents as having an unfair prior advantage over cliff-jumping agents, by virtue of their unfair refusal to jump off cliffs.  At this point you have covertly redefined "winning" as conformance to a particular ritual of cognition.  Pay attention to the money!

    Or here's another way of looking at it:  Faced with Newcomb's Problem, would you want to look really hard for a reason to believe that it was perfectly reasonable and rational to take only box B; because, if such a line of argument existed, you would take only box B and find it full of money?  Would you spend an extra hour thinking it through, if you were confident that, at the end of the hour, you would be able to convince yourself that box B was the rational choice?  This too is a rather odd position to be in.  Ordinarily, the work of rationality goes into figuring out which choice is the best - not finding a reason to believe that a particular choice is the best.

    Maybe it's too easy to say that you "ought to" two-box on Newcomb's Problem, that this is the "reasonable" thing to do, so long as the money isn't actually in front of you.  Maybe you're just numb to philosophical dilemmas, at this point.  What if your daughter had a 90% fatal disease, and box A contained a serum with a 20% chance of curing her, and box B might contain a serum with a 95% chance of curing her?  What if there was an asteroid rushing toward Earth, and box A contained an asteroid deflector that worked 10% of the time, and box B might contain an asteroid deflector that worked 100% of the time?

    Would you, at that point, find yourself tempted to make an unreasonable choice?

    If the stake in box B was something you could not leave behind?  Something overwhelmingly more important to you than being reasonable?  If you absolutely had to win - really win, not just be defined as winning?

    Would you wish with all your power that the "reasonable" decision was to take only box B?

    Then maybe it's time to update your definition of reasonableness.

    Alleged rationalists should not find themselves envying the mere decisions of alleged nonrationalists, because your decision can be whatever you like.  When you find yourself in a position like this, you shouldn't chide the other person for failing to conform to your concepts of reasonableness.  You should realize you got the Way wrong.

    So, too, if you ever find yourself keeping separate track of the "reasonable" belief, versus the belief that seems likely to be actually true.  Either you have misunderstood reasonableness, or your second intuition is just wrong.

    Now one can't simultaneously define "rationality" as the winning Way, and define "rationality" as Bayesian probability theory and decision theory.  But it is the argument that I am putting forth, and the moral of my advice to Trust In Bayes, that the laws governing winning have indeed proven to be math.  If it ever turns out that Bayes fails - receives systematically lower rewards on some problem, relative to a superior alternative, in virtue of its mere decisions - then Bayes has to go out the window.  "Rationality" is just the label I use for my beliefs about the winning Way - the Way of the agent smiling from on top of the giant heap of utility.  Currently, that label refers to Bayescraft.

    I realize that this is not a knockdown criticism of causal decision theory - that would take the actual book and/or PhD thesis - but I hope it illustrates some of my underlying attitude toward this notion of "rationality".

    You shouldn't find yourself distinguishing the winning choice from the reasonable choice.  Nor should you find yourself distinguishing the reasonable belief from the belief that is most likely to be true.

    That is why I use the word "rational" to denote my beliefs about accuracy and winning - not to denote verbal reasoning, or strategies which yield certain success, or that which is logically provable, or that which is publicly demonstrable, or that which is reasonable.

    As Miyamoto Musashi said:

    "The primary thing when you take a sword in your hands is your intention to cut the enemy, whatever the means. Whenever you parry, hit, spring, strike or touch the enemy's cutting sword, you must cut the enemy in the same movement. It is essential to attain this. If you think only of hitting, springing, striking or touching the enemy, you will not be able actually to cut him."


    New Comment
    616 comments, sorted by Click to highlight new comments since: Today at 6:08 AM
    Some comments are truncated due to high volume. (⌘F to expand all)Change truncation settings

    Either box B is already full or already empty.

    I'm not going to go into the whole literature, but the dominant consensus in modern decision theory is that one should two-box, and Omega is just rewarding agents with irrational dispositions. This dominant view goes by the name of "causal decision theory".

    I suppose causal decision theory assumes causality only works in one temporal direction. Confronted with a predictor that was right 100 out of 100 times, I would think it very likely that backward-in-time causation exists, and take only B. I assume this would, as you say, produce absurd results elsewhere.

    Decisions aren't physical.

    The above statement is at least hard to defend. Your decisions are physical and occur inside of you... So these two-boxers are using the wrong model amongst these two (see the drawings....)

    If you are a part of physics, so is your decision, so it must account for the correlation between your thought processes and the superintelligence. Once it accounts for that, you decide to one box, because you understood the entanglement of the computation done by omega and the physical process going inside your skull.

    If the entanglement is there, you are not looking at it from the outside, you are inside the process.

    Our minds have this quirk that makes us think there are two moments, you decide, and then you cheat, you get to decide again. But if you are only allowed to decide once, which is the case, you are rational by one-boxing.

    I think you capture the essence of the solution, here.
    1Motor Vehicle5mo
    Is it possible for someone to explain why, if your decision is a part of physics, your decision must account for the correlation between thought processes and the superintelligence? 
    From what I understand, to be a "Rational Agent" in game theory means someone who maximises their utility function (and not the one you ascribe to them). To say Omega is rewarding irrational agents isn't necessarily fair, since payoffs aren't always about the money. Lottery tickets are a good example this. What if my utility function says the worst outcome is living the rest of my life with regrets that I didn't one box? Then I can one box and still be a completely rational agent.

    You're complicating the problem too much by bringing in issues like regret. Assume for sake of argument that Newcomb's problem is to maximize the amount of money you receive. Don't think about extraneous utility issues.

    Fair point. There are too many hidden variables already without me explicitly adding more. If Newcomb's problem is to maximise money recieved (with no regard for what it seen as reasonable), the "Why ain't you rich argument seems like a fairly compelling one doesn't it? Winning the money is all that matters. I just realised that all I've really done is paraphrase the original post. Curse you source monitoring error!
    Lottery tickets exploit a completely different failure of rationality, that being our difficulties with small probabilities and big numbers, and our problems dealing with scale more generally. (ETA: The fantasies commonly cited in the context of lotteries' "true value" are a symptom of this failure.) It's not hard to come up with a game-theoretic agent that maximizes its payoffs against that kind of math. Second-guessing other agents' models is considerably harder. I haven't given much thought to this particular problem for a while, but my impression is that Newcomb exposes an exploit in simpler decision theories that's related to that kind of recursive modeling: naively, if you trust Omega's judgment of your psychology, you pick the one-box option, and if you don't, you pick up both boxes. Omega's track record gives us an excellent reason to trust its judgment from a probabilistic perspective, but it's trickier to come up with an algorithm that stabilizes on that solution without immediately trying to outdo itself.
    So for my own clarification, if I buy a lottery ticket with a perfect knowledge of how probable it is my ticket will win, does this make me irrational?
    Well, I fail to see any need for backward-in-time causation to get the prediction right 100 out of 100 times. As far as I understand, similar experiments have been performed in practice and homo sapiens are quite split in two groups 'one-boxers' and 'two-boxers' who generally have strong preferences towards one or other due to whatever differences in their education, logic experience, genetics, reasoning style or whatever factors that are somewhat stable specific to that individual. Having perfect predictive power (or even the possibility of it existing) is implied and suggested, but it's not really given, it's not really necessary, and IMHO it's not possible and not useful to use this 'perfect predictive power' in any reasoning here. From the given data in the situation (100 out of 100 that you saw), you know that Omega is a super-intelligent sorter who somehow manages to achieve 99.5% or better accuracy in sorting people into one-boxers and two-boxers. This accuracy seems also higher than the accuracy of most (all?) people in self-evaluation, i.e., as in many other decision scenarios, there is a significant difference in what people believe they would decide in situation X, and what they actually decide if it happens. [citation might be needed, but I don't have one at the moment, I do recall reading papers about such experiments]. The 'everybody is a perfect logician/rationalist and behaves as such' assumption often doesn't hold up in real life even for self-described perfect rationalists who make strong conscious effort to do so. In effect, data suggests that probably Omega knows your traits and decision chances (taking into account you taking into account all this) better than you do - it's simply smarter than homo sapiens. Assuming that this is really so, it's better for you to choose option B. Assuming that this is not so, and you believe that you can out-analyze Omega's perception of yourself, then you should choose the opposite of whatever Omega would t
    So what you're saying is that the only reason this problem is a problem is because the problem hasn't been defined narrowly enough. You don't know what Omega is capable of, so you don't know which choice to make. So there is no way to logically solve the problem (with the goal of maximizing utility) without additional information. Here's what I'd do: I'd pick up B, open it, and take A iff I found it empty. That way, Omega's decision of what to put in the box would have to incorporate the variable of what Omega put in the box, causing an infinite regress which will use all cpu cycles until the process is terminated. Although that'll probably result in the AI picking an easier victim to torment and not even giving me a measly thousand dollars.
    Okay... so since you already know, in advance of getting the boxes, that that's what you'd know, Omega can deduce that. So you open Box B, find it empty, and then take Box A. Enjoy your $1000. Omega doesn't need to infinite loop that one; he knows that you're the kind of person who'd try for Box A too.
    No, putting $1 million in box B works to. Origin64 opens box B, takes the money, and doesn't take box A. It's like "This sentence is true." - whatever Omega does makes the prediction valid.
    Which means you might end up with either amount of money, since you don't really know enough about Omega , instead of just the one box winnings. So you should still just one box?
    Not how Omega looks at it. By definition, Omega looks ahead, sees a branch in which you would go for Box A, and puts nothing in Box B. There's no cheating Omega... just like you can't think "I'm going to one-box, but then open Box A after I've pocketed the million" there's no "I'm going to open Box B first, and decide whether or not to open Box A afterward". Unless Omega is quite sure that you have precommitted to never opening Box A ever, Box B contains nothing; the strategy of leaving Box A as a possibility if Box B doesn't pan out is a two-box strategy, and Omega doesn't allow it.
    Well, this isn't quite true. What Omega cares about is whether you will open Box A. From Omega's perspective it makes no difference whether you've precommitted to never opening it, or whether you've made no such precommitment but it turns out you won't open it for other reasons.
    Assuming that Omega's "prediction" is in good faith, and that we can't "break" him as a predictor as a side effect of exploiting casuality loops etc. in order to win.
    I'm not sure I understood that, but if I did, then yes, assuming that Omega is as described in the thought experiment. Of course, if Omega has other properties (for example, is an unreliable predictor) other things follow.
    If you look in box B before deciding whether to choose box A, then you can force Omega to be wrong. That sounds like so much fun that I might choose it over the $1000.
    That's the popular understanding (or lack thereof) here and among philosophers in general. Philosophers just don't get math. If the decision theory is called causal but doesn't itself make any references to physics, then that's a slightly misleading name. I've written on that before The math doesn't go "hey hey, the theory is named causal therefore you can't treat 2 robot arms controlled by 2 control computers that run one function on one state, the same as 2 robot arms controlled by 1 computer". Confused sloppy philosophers do. Also, the best case is to be predicted to 1-box but 2-box in reality. If the prediction works by backwards causality, well then causal decision theory one-boxes. If the prediction works by simulation, the causal decision theory can either have world model where both the value inside predictor and the value inside actual robot are represented by same action A, and 1-box, or it can have uncertainty as of whenever the world outside of it is normal reality or predictor's simulator, where it will again one box (assuming it cares about the real money even if it is inside predictor, which it would if it needs money to pay for e.g. it's child's education). It will also 1-box in simulator and 2-box in reality if it can tell those apart.
    I'm confused. Causal decision theory was invented or formalised almost entirely by philosophers. It takes the 'causal' in its name from its reliance on inductive logic and inference. It doesn't make sense to claim that philosophers are being sloppy about the word 'causal' here, and claiming that causal decision theory will accept backwards causality and one-box is patently false unless you mean something other than what the symbol 'causal decision theory' refers to when you say 'causal decision theory'.
    Firstly, the notion that the actions should be chosen based on their consequences, taking the actions as cause of the consequences, was definitely not invented by philosophers. Secondarily, the logical causality is not identical to physical causality (the latter is dependent on specific laws of physics). Thirdly, not all philosophers are sloppy; some are very sloppy some are less sloppy. Fourth, anything that was not put in mathematical form to be manipulated using formal methods, is not formalized. When you formalize stuff you end up stripping notion of self unless explicitly included as part of formalism, stripping notion of the time where the math is working unless explicitly included as part of formalism, and so on, ending up without the problem. Maybe you are correct; it is better to let symbol 'causal decision theory' to refer to confused philosophy. Then we would need some extra symbol for how the agents implementable using mathematics actually decide (and how robots that predict outcomes of their actions on a world model actually work), which is very very similar to 'causal decision theory' sans all the human preconditions of what self is.
    I notice I actually agree with you - if we did try, using mathematics, to implement agents who decide and predict in the manner you describe, we'd find it incorrect to describe these agents as causal decision theory agents. In fact, I also expect we'd find ourselves disillusioned with CDT in general, and if philosophers brought it up, we'd direct them to instead engage with the much more interesting agents we've mathematically formalised.
    Well, each philosopher's understanding of CDT seem to differ from the other: The notion that the actions should be chosen based on consequences - as expressed in the formula here - is perfectly fine, albeit incredibly trivial. Can formalize that all the way into agent. Written such agents myself. Still need a symbol to describe this type of agent. But philosophers go from this to "my actions should be chosen based on consequences", and it is all about the true meaning of self and falls within the purview of your conundrums of philosophy . Having 1 computer control 2 robots arms wired in parallel, and having 2 computers running exact same software as before, controlling 2 robot arms, there's no difference for software engineering, its a minor detail that has been entirely abstracted from software. There is difference for philosophizing thought because you can't collapse logical consequences and physical causality into one thing in the latter case. edit: anyhow. to summarize my point: In terms of agents actually formalized in software, one-boxing is only a matter of implementing predictor into world model somehow, either as second servo controlled by same control variables, or as uncertain world state outside the senses (in the unseen there's either real world or simulator that affects real world via hand of predictor). No conceptual problems what so ever. edit: Good analogy, 'twin paradox' in special relativity. There's only paradox if nobody done the math right.
    @Nick_Tarleton Agreed, the problem immediately reminded me of "retroactive preparation" and time-loop logic. It is not really the same reasonning, but it has the same "turn causality on its head" aspect. If I don't have proof of the reliability of Omega's predictions, I find myself less likely to be "unreasonnable" when the stakes are higher (that is, I'm more likely to two-box if it's about saving the world). I find it highly unlikely that an entity wandering across worlds can predict my actions to this level of detail, as it seems way harder than traveling through space or teleporting money. I might risk a net loss of $1 000 to figure it out (much like I'd be willing to spend $1000 to interact with such a space-traveling stuff-teleporting entity), but not a loss of a thousand lives. In the game as the article describe it, I would only one-box if "the loss of what box A contains and nothing in B" was an acceptable outcome. I would be increasingly likely to one-box as the probability of the AI being actually able to predict my actions in advance increases.
    The thing is, this 'modern decision theory', rather than being some sort of central pillar as you'd assume from the name, is mostly philosophers "struggling in the periphery to try to tell us something", as Feynman once said about philosophers of science. When it comes to any actual software which does something, this everyday notion of 'causality' proves to be a very slippery concept. This Rude Goldberg machine - like model of the world, where you push a domino and it pushes another domino, and the chain goes to your reward, that's just very approximate physics that people tend to use to make decisions, it's not fundamental, and interesting models of decision making are generally set up to learn that from observed data (which of course makes it impossible to do lazy philosophy involving various verbal hypotheticals where the observations that would lead the agent to believe the problem set up are not specified).

    People seem to have pretty strong opinions about Newcomb's Problem. I don't have any trouble believing that a superintelligence could scan you and predict your reaction with 99.5% accuracy.

    I mean, a superintelligence would have no trouble at all predicting that I would one-box... even if I hadn't encountered the problem before, I suspect.

    Ultimately you either interpret "superintelligence" as being sufficient to predict your reaction with significant accuracy, or not. If not, the problem is just a straightforward probability question, as explained here, and becomes uninteresting.

    Otherwise, if you interpret "superintelligence" as being sufficient to predict your reaction with significant accuracy (especially a high accuracy like >99.5%), the words of this sentence...

    And the twist is that Omega has put a million dollars in box B iff Omega has predicted that you will take only box B.

    ...simply mean "One-box to win, with high confidence."

    Summary: After disambiguating "superintelligence" (making the belief that Omega is a superintelligence pay rent), Newcomb's problem turns into either a straightforward probability question or a fairly simple issue of rearranging the words in equivalent ways to make the winning answer readily apparent.

    If you won't explicitly state your analysis, maybe we can try 20 questions?

    I have suspected that supposed "paradoxes" of evidential decision theory occur because not all the evidence was considered. For example, the fact that you are using evidential decision theory to make the decision.


    Hmm, changed my mind, should have thought more before writing... the EDT virus has early symptoms of causing people to use EDT before progressing to terrible illness and death. It seems EDT would then recommend not using EDT.

    I one-box, without a moment's thought.

    The "rationalist" says "Omega has already left. How could you think that your decision now affects what's in the box? You're basing your decision on the illusion that you have free will, when in fact you have no such thing."

    To which I respond "How does that make this different from any other decision I'll make today?"

    I think the two box person is confused about what it is to be rational, it does not mean "make a fancy argument," it means start with the facts, abstract from them, and reason about your abstractions.

    In this case if you start with the facts you see that 100% of people who take only box B win big, so rationally, you do the same. Why would anyone be surprised that reason divorced from facts gives the wrong answer?

    Precisely. I've been reading a lot about the Monty Hall problem recently, and I feel that it's a relevant conundrum. The confused rationalist will say: but my choice CANNOT cause a linear entaglement, the reward is predecided. But the functional rationalist will see that agents who one-box (or switch doors, in the case of Monty Hall) consistently win. It is demonstrably a more effective strategy. You work with the facts and evidence available to you. Regardless of how counter-intuitive the resulting strategy becomes.
    Precisely. I've been reading a lot about the Monty Hall Problem recently (, and I feel that it's a relevant conundrum. The confused rationalist will say: but my choice CANNOT cause a linear entaglement, the reward is predecided. But the functional rationalist will see that agents who one-box (or switch doors, in the case of Monty Hall) consistently win. It is demonstrably a more effective strategy. You work with the facts and evidence available to you and abstract out from there. Regardless of how counter-intuitive the resulting strategy becomes.

    This dilemma seems like it can be reduced to:

    1. If you take both boxes, you will get $1000
    2. If you only take box B, you will get $1M Which is a rather easy decision.

    There's a seemingly-impossible but vital premise, namely, that your action was already known before you acted. Even if this is completely impossible, it's a premise, so there's no point arguing it.

    Another way of thinking of it is that, when someone says, "The boxes are already there, so your decision cannot affect what's in them," he is wrong. It has been assumed that your decision does affect what's in them, so the fact that you cannot imagine how that is possible is wholly irrelevant.

    In short, I don't understand how this is controversial when the decider has all the information that was provided.

    Actually, we don't know that our decision affects the contents of Box B. In fact, we're told that it contains a million dollars if-and-only-if Omega predicts we will only take Box B. It is possible that we could pick Box B even tho Omega predicted we would take both boxes. Omega has only observed to have predicted correctly 100 times. And if we are sufficiently doubtful whether Omega would predict that we would take only Box B, it would be rational to take both boxes. Only if we're somewhat confident of Omega's prediction can we confidently one-box and rationally expect it to contain a million dollars.
    51% confidence would suffice. * Two-box expected value: 0.51 $1K + 0.49 $1.001M = $491000 * One-box expected value: 0.51 $1M + 0.49 $0 = $510000
    You're saying that we live in a universe where Newcomb's problem is impossible because the future doesn't effect the past. I'll re-phrase this problem in such a way that it seems plausible in our universe: I've got really nice scanning software. I scan your brain down to the molecule, and make a virtual representation of it on a computer. I run virtual-you in my software, and give virtual-you Newcomb's problem. Virtual-you answers, and I arrange my boxes according to that answer. I come back to real-you. You've got no idea what's going on. I explain the scenario to you and I give you Newcomb's problem. How do you answer? This particular instance of the problem does have an obvious, relatively uncomplicated solution: Lbh unir ab jnl bs xabjvat jurgure lbh ner rkcrevrapvat gur cneg bs gur fvzhyngvba, be gur cneg bs gur syrfu-naq-oybbq irefvba. Fvapr lbh xabj gung obgu jvyy npg vqragvpnyyl, bar-obkvat vf gur fhcrevbe bcgvba. If for any reason you suspect that the Predictor can reach a sufficient level of accuracy to justify one-boxing, you one box. It doesn't matter what sort of universe you are in.
    Not that I disagree with the one-boxing conclusion, but this formulation requires physically reducible free will (which has recently been brought back into discussion). It would also require knowing the position and momentum of a lot of particles to arbitrary precision, which is provably impossible.
    We don't need a perfect simulation for the purposes of this problem in the abstract - we just need a situation such that the problem-solver assigns better-than-chance predicting power to the Predictor, and a sufficiently high utility differential between winning and losing. The "perfect whole brain simulation" is an extreme case which keeps things intuitively clear. I'd argue that any form of simulation which performs better than chance follows the same logic. The only way to escape the conclusion via simulation is if you know something that Omega doesn't - for example, you might have some secret external factor modify your "source code" and alter your decision after Omega has finished examining you. Beating Omega essentially means that you need to keep your brain-state in such a form that Omega can't deduce that you'll two-box. As Psychohistorian3 pointed out, the power that you've assigned to Omega predicting accurately is built into the problem. Your estimate of the probability that you will succeed in deception via the aforementioned method or any other is fixed by the problem. In the real world, you are free to assign whatever probability you want to your ability to deceive Omega's predictive mechanisms, which is why this problem is counter intuitive.
    7Eliezer Yudkowsky10y
    Also: You can't simultaneously claim that any rational being ought to two-box, this being the obvious and overdetermined answer, and also claim that it's impossible for anyone to figure out that you're going to two-box.
    Right, any predictor with at least a 50.05% accuracy is worth one-boxing upon (well, maybe a higher percentage for those with concave functions in money). A predictor with sufficiently high accuracy that it's worth one-boxing isn't unrealistic or counterintuitive at all in itself, but it seems (to me at least) that many people reach the right answer for the wrong reason: the "you don't know whether you're real or a simulation" argument. Realistically, while backwards causality isn't feasible, neither is precise mind duplication. The decision to one-box can be rationally reached without those reasons: you choose to be the kind of person to (predictably) one-box, and as a consequence of that, you actually do one-box.
    Oh, that's fair. I was thinking of "you don't know whether you're real or a simulation" as an intuitive way to prove the case for all "conscious" simulations. It doesn't have to be perfect - you could just as easily be an inaccurate simulation, with no way to know that you are a simulation and no way to know that you are inaccurate with respect to an original. I was trying to get people to generalize downwards from the extreme intuitive example- Even with decreasing accuracy, as the simulation becomes so rough as to lose "consciousness" and "personhood", the argument keeps holding.
    Yeah, the argument would hold just as much with an inaccurate simulation as with an accurate one. The point I was trying to make wasn't so much that the simulation isn't going to be accurate enough, but that a simulation argument shouldn't be a prerequisite to one-boxing. If the experiment were performed with human predictors (let's say a psychologist who predicts correctly 75% of the time), one-boxing would still be rational despite knowing you're not a simulation. I think LW relies on computationalism as a substitute for actually being reflectively consistent in problems such as these.
    The trouble with real world examples is that we start introducing knowledge into the problem that we wouldn't ideally have. The psychologist's 75% success rate doesn't necessarily apply to you - in the real world you can make a different estimate than the one that is given. If you're an actor or a poker player, you'll have a much different estimate of how things are going to work out. Psychologists are just messier versions of brain scanners - the fundamental premise is that they are trying to access your source code. And what's more - suppose the predictions weren't made by accessing your source code? The direction of causality does matter. If Omega can predict the future, the causal lines flow backwards from your choice to Omega's past move. If Omega is scanning your brain, the causal lines go from your brain-state to Omega's decision. If there are no causal lines between your brain/actions and Omega's choice, you always two-box. Real world example: what if I substituted your psychologist for a sociologist, who predicted you with above-chance accuracy using only your demographic factors? In this scenario, you aught to two-box - If you disagree, let me know and I can explain myself. In the real world, you don't know to what extent your psychologist is using sociology (or some other factor outside your control). People can't always articulate why, but their intuition (correctly) begins to make them deviate from the given success% estimate as more of these real-world variables get introduced.
    True, the 75% would merely be a past history (and I am in fact a poker player). Indeed, if the factors used were entirely or mostly comprised of factors beyond my control (and I knew this), I would two-box. However, two-boxing is not necessarily optimal because of a predictor whose prediction methods you do not know the mechanics of. In the limited predictor problem, the predictor doesn't use simulations/scanners of any sort but instead uses logic, and yet one-boxers still win.
    agreed. To add on to this: It's worth pointing out that Newcomb's problem always takes the form of Simpson's paradox. The one boxers beat the two boxers as a whole, but among agents predicted to one-box, the two boxers win, and among agents predicted to two-box, the two boxers win. The only reason to one-box is when your actions (which include both the final decision and the thoughts leading up to it) effect Omega's prediction. The general rule is: "Try to make Omega think you're one-boxing, but two-box whenever possible." It's just that in Newcomb's problem proper, fulfilling the first imperative requires actually one-boxing.
    So you would never one-box unless the simulator did some sort of scan/simulation upon your brain? But it's better to one-box and be derivable as the kind of person to (probably) one-box than to two-box and be derivable as the kind of person to (probably) two-box. Your final decision never affects the actual arrangement of the boxes, but its causes do.
    I'd one-box when Omega had sufficient access to my source-code. It doesn't have to be through scanning - Omega might just be a great face-reading psychologist. We're in agreement. As we discussed, this only applies insofar as you can control the factors that lead you to be classified as a one-boxer or a two-boxer. You can alter neither demographic information nor past behavior. But when (and only when) one-boxing causes you to be derived as a one-boxer, you should obviously one box. Well, that's true for this universe. I just assume we're playing in any given universe, some of which include Omegas who can tell the future (which implies bidirectional causality) - since Psychohistorian3 started out with that sort of thought when I first commented.
    Ok, so we do agree that it can be rational to one-box when predicted by a human (if they predict based upon factors you control such as your facial cues). This may have been a misunderstanding between us then, because I thought you were defending the computationalist view that you should only one-box if you might be an alternate you used in the prediction.
    yes, we do agree on that.
    Assuming that you have no information other than the base rate, and that it's equally likely to be wrong either way.
    An alternate solution which results in even more winning is to cerqvpg gung V znl or va fhpu n fvghngvba va gur shgher. Unir n ubbqyhz cebzvfr gung vs V'z rire va n arjpbzoyvxr fvghngvba gung ur jvyy guerngra gb oernx zl yrtf vs V qba'g 2-obk. Cnl gur ubbqyhz $500 gb frpher uvf cebzvfr. Gura pbzcyrgryl sbetrg nobhg gur jubyr neenatrzrag naq orpbzr n bar-obkre. Fpnaavat fbsgjner jvyy cerqvpg gung V 1-obk, ohg VEY V'z tbvat gb 2-obk gb nibvq zl yrtf trggvat oebxra.
    1Marion Z.10mo
    But you've perfectly forgotten about the hoodlum, so you will in fact one box. Or, does the hoodlum somehow show up and threaten you in the moment between the scanner filling the boxes and you making your decision? That seems to add an element of delay and environmental modification that I don't think exists in the original problem, unless I'm misinterpreting.  Also, I feel like by analyzing your brain to some arbitrarily precise standard, the scanner could see 3 things:  You are (or were at some point in the past) likely to think of this solution, you are/were likely to actually go through with this solution, and the hoodlum's threat would, in fact, cause you to two-box, letting the scanner predict that you will two-box.
    Your decision doesn't affect what's in the boxes, but your decision procedure does, and that already exists when the question's being assigned. It may or may not be possible to derive your decision from the decision procedure you're using in the general case -- I haven't actually done the reduction, but at first glance it looks cognate to some problems that I know are undecidable -- but it's clearly possible in some cases, and it's at least not completely absurd to imagine an Omega with a very high success rate. As best I can tell, most of the confusion here comes from a conception of free will that decouples the decision from the procedure leading to it.
    Yeah, agreed. I often describe this as NP being more about what kind of person I am than it is about what decision I make, but I like your phrasing better.

    I'd love to say I'd find some way of picking randomly just to piss Omega off, but I'd probably just one-box it. A million bucks is a lot of money.

    2Ramana Kumar14y
    Would that make you a supersuperintelligence? Since I presume by "picking randomly" you mean randomly to Omega, in other words Omega cannot find and process enough information to predict you well. Otherwise what does "picking randomly" mean?
    The definition of omega as something that can predict your actions leads it to have some weird powers. You could pick a box based on the outcome of a quantum event with a 50% chance, then omega would have to vanish in a puff of physical implausibility.
    1Ramana Kumar14y
    What's wrong with Omega predicting a "quantum event"? "50% chance" is not an objective statement, and it may well be that Omega can predict quantum events. (If not, can you explain why not, or refer me to an explanation?)
    From wikipedia "In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space.[9] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments." This is the best formalism we have for predicting things at this scale and it only spits out probabilities. I would be surprised if something did a lot better!
    0Ramana Kumar14y
    As I understand it, probabilities are observed because there are observers in two different amplitude blobs of configuration space (to use the language of the quantum physics sequence) but "the one we are in" appears to be random to us. And mathematically I think quantum mechanics is the same under this view in which there is no "inherent, physical" randomness (so it would still be the best formalism we have for predicting things). Could you say what "physical randomness" could be if we don't allow reference to quantum mechanics? (i.e. is that the only example? and more to the point, does the notion make any sense?)
    You seem to have transitioned to another argument here... please clarify what this has to do with omega and its ability to predict your actions.
    0Ramana Kumar14y
    The new argument is about whether there might be inherently unpredictable things. If not, then your picking a box based on the outcome of a "quantum event" shouldn't make Omega any less physically plausible,
    What I didn't understand is why you removed quantum experiments from the discussion. I believe it is very plausible to have something that is physically unpredictable, as long as the thing doing the predicting is bound by the same laws as what you are trying to predict. Consider a world made of reversible binary gates with the same number of inputs as outputs (that is every input has a unique output, and vice versa). We want to predict one complex gate. Not a problem, just clone all the inputs and copy the gate. However you have to do that only using reversible binary gates. Lets start with cloning the bits. In is what you are trying to copy without modifying so that you can predict what affect it will have on the rest of the system. You need a minimum of two outputs, so you need another input B. You get to create the gate in order to copy the bit and predict the system. The ideal truth table looks something like In | B | Out | Copy 0 | 0 | 0 | 0 0 | 1 | 0 | 0 1 | 0 | 1 | 1 1 | 1 | 1 | 1 This violates our reversibility assumption. The best copier we could make is In | B | Out | Copy 0 | 0 | 0 | 0 0 | 1 | 1 | 0 1 | 0 | 0 | 1 1 | 1 | 1 | 1 This copies precisely, but mucks up the output making our copy useless for prediction. If you could control B, or knew the value of B then we could correct the Output. But as I have shown here finding out the value of a bit is non-trivial. The best we could do would be to find sources of bits with statistically predictable properties then use them for duplicating other bits. The world is expected to be reversible, and the no cloning theorem applies to reality which I think is stricter than my example. However I hope I have shown how a simple lawful universe can be hard to predict by something inside it. In short, stop thinking of yourself (and Omega) as an observer outside physics that does not interact with the world. Copying is disturbing.
    Even though I do not have time to reflect on the attempted proof and even though the attempted proof is best described as a stab at a sketch of a proof and even though this "reversible logic gates" approach to a proof probably cannot be turned into an actual proof and even though Nick Tarleton just explained why the "one box or two box depending on an inherently unpredictable event" strategy is not particularly relevant to Newcomb's, I voted this up and I congratulate the author (whpearson) because it is an attempt at an original proof of something very cool (namely, limits to an agent's ability to learn about its environment) and IMHO probably relevant to the Friendliness project. More proofs and informed stabs at proofs, please!
    I suspect Omega would know you were going to do that, and would be able to put the box in a superposition dependent on the same quantum event, so that in the branches where you 1-box, box B contains $1million, and where you 2-box it's empty.
    Exactly what I was thinking.

    It's often stipulated that if Omega predicts you'll use some randomizer it can't predict, it'll punish you by acting as if it predicted two-boxing.

    (And the most favourable plausible outcome for randomizing would be scaling the payoff appropriately to the probability assigned.)
    Newcomb's problem doesn't specify how Omega chooses the 'customers'. It's a quite realistic possibility that it simply has not offered the choice to anyone that would use a randomizer, and cherrypicked only the people which have at least 99.9% 'prediction strength'.

    It's a great puzzle. I guess this thread will degenerate into arguments pro and con. I used to think I'd take one box, but I read Joyce's book and that changed my mind.

    For the take-one-boxers:

    Do you believe, as you sit there with the two boxes in front of you, that their contents are fixed? That there is a "fact of the matter" as to whether box B is empty or not? Or is box B in a sort of intermediate state, halfway between empty and full? If so, do you generally consider that things momentarily out of sight may literally change their physical sta... (read more)

    Na-na-na-na-na-na, I am so sorry you only got $1000!

    Me, I'm gonna replace my macbook pro, buy an apartment and a car and take a two week vacation in the Bahamas, and put the rest in savings!


    Point: arguments don't matter, winning does.

    Oops. I had replied to this until I saw its parent was nearly 3 years old. So as I don't (quite) waste the typing:

    Do you believe, as you sit there with the two boxes in front of you, that their contents are fixed?


    That there is a "fact of the matter" as to whether box B is empty or not?


    Or is box B in a sort of intermediate state, halfway between empty and full?


    If so, do you generally consider that things momentarily out of sight may literally change their physical states into something indeterminate?


    Do you picture box B literally becoming empty and full as you change your opinion back and forth?

    If not, if you think box B is definitely either full or empty and there is no unusual physical state describing the contents of that box, then would you agree that nothing you do now can change the contents of the box?


    And if so, then taking the additional box cannot reduce what you get in box B.

    No, it can't. (But it already did.)

    If I take both boxes how much money do I get? $1,000

    If I take one box how much money do I get? $10,000,000 (or whatever it was instantiated to.)

    It seems that my questions were more useful than yours. Perhaps Joyce b... (read more)

    Yes. Yes. No. No. Yes. No, it can't. (But it already did.) If I take both boxes how much money do I get? $1,000 If I take one box how much money do I get? $10,000,000 (or whatever it was instantiated to.) It seems that my questions were more useful than yours. Perhaps Joyce beffudled you? It could be that he missed something. (Apart from counter-factual $9,999,000.) I responded to all your questions with the answers you intended to make the point that I don't believe those responses are at all incompatible with making the decision that earns you lots and lots of money.

    To quote E.T. Jaynes:

    "This example shows also that the major premise, “If A then B” expresses B only as a logical consequence of A; and not necessarily a causal physical consequence, which could be effective only at a later time. The rain at 10 AM is not the physical cause of the clouds at 9:45 AM. Nevertheless, the proper logical connection is not in the uncertain causal direction (clouds =⇒ rain), but rather (rain =⇒ clouds) which is certain, although noncausal. We emphasize at the outset that we are concerned here with logical connections, because some discussions and applications of inference have fallen into serious error through failure to see the distinction between logical implication and physical causation. The distinction is analyzed in some depth by H. A. Simon and N. Rescher (1966), who note that all attempts to interpret implication as expressing physical causation founder on the lack of contraposition expressed by the second syllogism (1–2). That is, if we tried to interpret the major premise as “A is the physical cause of B,” then we would hardly be able to accept that “not-B is the physical cause of not-A.” In Chapter 3 we shall see that attempts to interpret plausible inferences in terms of physical causation fare no better."

    @: Hal Finney:

    Certainly the box is either full or empty. But the only way to get the money in the hidden box is to precommit to taking only that one box. Not pretend to precommit, really precommit. If you try to take the $1,000, well then I guess you really hadn't precommitted after all. I might vascillate, I might even be unable to make such a rigid precommitment with myself (though I suspect I am), but it seems hard to argue that taking only one box is not the correct choice.

    I'm not entirely certain that acting rationally in this situation doesn't require an element of doublethink, but thats a topic for another post.

    I would be interested in know if your opinion would change if the "predictions" of the super-being were wrong .5% of the time, and some small number of people ended up with the $1,001,000 and some ended up with nothing. Would you still 1 box it?

    If a bunch of people have played the game already, then you can calculate the average payoff for a 1-boxer and that of a 2-boxer and pick the best one.

    I suppose I might still be missing something, but this still seems to me just a simple example of time inconsistency, where you'd like to commit ahead of time to something that later you'd like to violate if you could. You want to commit to taking the one box, but you also want to take the two boxes later if you could. A more familiar example is that we'd like to commit ahead of time to spending effort to punish people who hurt us, but after they hurt us we'd rather avoid spending that effort as the harm is already done.

    If I know that the situation has resolved itself in a manner consistent with the hypothesis that Omega has successfully predicted people's actions many times over, I have a high expectation that it will do so again.

    In that case, what I will find in the boxes is not independent of my choice, but dependent on it. By choosing to take two boxes, I cause there to be only $1,000 there. By choosing to take only one, I cause there to be $1,000,000. I can create either condition by choosing one way or another. If I can select between the possibilities, I prefer... (read more)

    Prediction <-> our choice, if we use the 100/100 record as equivalent with complete predictive accuracy. The "weird thing going on here" is that one value is set (that's what "he has already flown away" does), yet we are being told that we can change the other value. You see these reactions: 1) No, we can't toggle the other value, actually. Choice is not really in the premise, or is breaking the premise. 2) We can toggle the choice value, and it will set the predictive value accordingly. The prior value of the prediction does not exist or is not relevant. We have already equated "B wins" with "prediction value = B" wlog. If we furthermore have equated "choice value = B" with "prediction value = B" wlog, we have two permissible arrays of values: all A, or all B. Now our knowledge is restricted to choice value. We can choose A or B. Since the "hidden" values are known to be identical to the visible value, we should pick the visible value in accordance with what we want for a given other value. Other thoughts: -Locally, it appears that you cannot "miss out" because within a value set, your choice value is the only possible one in identity with the other values. -This is a strange problem, because generally paradox provokes these kinds of responses. In this case, however, fixing a value does not cause a contradiction both ways. If you accept the premise and my premises above, there should be no threat of complications from Omega or anything else. -if 1 and 2 really are the only reactions, and 2 ->onebox, any twoboxers must believe 1. But this is absurd. So whence the twoboxers?

    I don't know the literature around Newcomb's problem very well, so excuse me if this is stupid. BUT: why not just reason as follows:

    1. If the superintelligence can predict your action, one of the following two things must be the case:

    a) the state of affairs whether you pick the box or not is already absolutely determined (i.e. we live in a fatalistic universe, at least with respect to your box-picking)

    b) your box picking is not determined, but it has backwards causal force, i.e. something is moving backwards through time.

    If a), then practical reason is ... (read more)


    Once we can model the probabilities of the various outcomes in a noncontroversial fashion, the specific choice to make depends on the utility of the various outcomes. $1,001,000 might be only marginally better than $1,000,000 -- or that extra $1,000 could have some significant extra utility.

    If we assume that Omega almost never makes a mistake and we allow the chooser to use true randomization (perhaps by using quantum physics) in making his choice, then Omega must make his decision in part through seeing into the future. In this case the chooser should obviously pick just B.

    Hanson: I suppose I might still be missing something, but this still seems to me just a simple example of time inconsistency

    In my motivations and in my decision theory, dynamic inconsistency is Always Wrong. Among other things, it always implies an agent unstable under reflection.

    A more familiar example is that we'd like to commit ahead of time to spending effort to punish people who hurt us, but after they hurt us we'd rather avoid spending that effort as the harm is already done.

    But a self-modifying agent would modify to not rather avoid it.

    Gowder: If... (read more)

    I don't see why this needs to be so drawn out.

    I know the rules of the game. I also know that Omega is super intelligent, namely, Omega will accurately predict my action. Since Omega knows that I know this, and since I know that he knows I know this, I can rationally take box B, content in my knowledge that Omega has predicted my action correctly.

    I don't think it's necessary to precommit to any ideas, since Omega knows that I'll be able to rationally deduce the winning action given the premise.

    We don't even need a superintelligence. We can probably predict on the basis of personality type a person's decision in this problem with an 80% accuracy, which is already sufficient that a rational person would choose only box B.

    The possibility of time inconsistency is very well established among game theorists, and is considered a problem of the game one is playing, rather than a failure to analyze the game well. So it seems you are disagreeing with most all game theorists in economics as well as most decision theorists in philosophy. Maybe perhaps they are right and you are wrong?

    The interesting thing about this game is that Omega has magical super-powers that allow him to know whether or not you will back out on your commitment ahead of time, and so you can make your commitment credible by not being going to back out on your commitment. If that makes any sense.

    Robin, remember I have to build a damn AI out of this theory, at some point. A self-modifying AI that begins anticipating dynamic inconsistency - that is, a conflict of preference with its own future self - will not stay in such a state for very long... did the game theorists and economists work a standard answer for what happens after that?

    If you like, you can think of me as defining the word "rationality" to refer to a different meaning - but I don't really have the option of using the standard theory, here, at least not for longer than 50 milliseconds.

    If there's some nonobvious way I could be wrong about this point, which seems to me quite straightforward, do let me know.

    In reality, either I am going to take one box or two. So when the two-boxer says, "If I take one box, I'll get amount x," and "If I take two boxes, I'll get amount x+1000," one of these statements is objectively counterfactual. Let's suppose he is going to in fact take both boxes. Then his second takement is factual and his first statement counterfactual. Then his two statements are:

    1)Although I am not in fact going to take only one box, were I to take only box, I would get amount x, namely the amount that would be in the box.

    2)I am in ... (read more)

    Eleizer: whether or not a fixed future poses a problem for morality is a hotly disputed question which even I don't want to touch. Fortunately, this problem is one that is pretty much wholly orthogonal to morality. :-)

    But I feel like in the present problem the fixed future issue is a key to dissolving the problem. So, assume the box decision is fixed. It need not be the case that the stress is fixed too. If the stress isn't fixed, then it can't be relevant to the box decision (the box is fixed regardless of your decision between stress and no-stress).... (read more)

    Paul, being fixed or not fixed has nothing to do with it. Suppose I program a deterministic AI to play the game (the AI picks a box.)

    The deterministic AI knows that it is deterministic, and it knows that I know too, since I programmed it. So I also know whether it will take one or both boxes, and it knows that I know this.

    At first, of course, it doesn't know itself whether it will take one or both boxes, since it hasn't completed running its code yet. So it says to itself, "Either I will take only one box or both boxes. If I take only one box, the pro... (read more)

    I practice historical European swordsmanship, and those Musashi quotes have a certain resonance to me*. Here is another (modern) saying common in my group:

    If it's stupid, but it works, then it ain't stupid.

    • you previously asked why you couldn't find similar quotes from European sources - I believe this is mainly a language barrier: The English were not nearly the swordsmen that the French, Italians, Spanish, and Germans were (though they were pretty mean with their fists). You should be able to find many quotes in those other languages.

    Eliezer, I don't read the main thrust of your post as being about Newcomb's problem per se. Having distinguished between 'rationality as means' to whatever end you choose, and 'rationality as a way of discriminating between ends', can we agree that the whole specks / torture debate was something of a red herring ? Red herring, because it was a discussion on using rationality to discriminate between ends, without having first defined one's meta-objectives, or, if one's meta-objectives involved hedonism, establishing the rules for performing math over subje... (read more)

    Unknown: your last question highlights the problem with your reasoning. It's idle to ask whether I'd go and jump off a cliff if I found my future were determined. What does that question even mean?

    Put a different way, why should we ask an "ought" question about events that are determined? If A will do X whether or not it is the case that a rational person will do X, why do we care whether or not it is the case that a rational person will do X? I submit that we care about rationality because we believe it'll give us traction on our problem of ... (read more)

    Paul, it sounds like you didn't understand. A chess playing computer program is completely deterministic, and yet it has to consider alternatives in order to make its move. So also we could be deterministic and we would still have to consider all the possibilities and their benefits before making a move.

    So it makes sense to ask whether you would jump off a cliff if you found out that the future is determined. You would find out that the future is determined without knowing exactly which future is determined, just like the chess program, and so you would ha... (read more)

    I do understand. My point is that we ought not to care whether we're going to consider all the possibilities and benefits.

    Oh, but you say, our caring about our consideration process is a determined part of the causal chain leading to our consideration process, and thus to the outcome.

    Oh, but I say, we ought not to care* about that caring. Again, recurse as needed. Nothing you can say about the fact that a cognition is in the causal chain leading to a state of affairs counts as a point against the claim that we ought not to care about whether or not we have that cognition if it's unavoidable.

    The paradox is designed to give your decision the practical effect of causing Box B to contain the money or not, without actually labeling this effect "causation." But I think that if Box B acts as though its contents are caused by your choice, then you should treat it as though they were. So I don't think the puzzle is really something deep; rather, it is a word game about what it means to cause something.

    Perhaps it would be useful to think about how Omega might be doing its prediction. For example, it might have the ability to travel into the f... (read more)

    I have two arguments for going for Box B. First, for a scientist it's not unusual that every rational argument (=theory) predicts that only two-boxing makes sense. Still, if the experiment again and again refutes that, it's obviously the theory that's wrong and there's obviously something more to reality than that which fueled the theories. Actually, we even see dilemmas like Newcomb's in the contextuality of quantum measurements. Measurement tops rationality or theory, every time. That's why science is successful and philosophy is not.

    Second, there's no q... (read more)

    Paul, if we were determined, what would you mean when you say that "we ought not to care"? Do you mean to say that the outcome would be better if we didn't care? The fact that the caring is part of the causal chain does have something to do with this: the outcome may be determined by whether or not we care. So if you consider one outcome better than another (only one really possible, but both possible as far as you know), then either "caring" or "not caring" might be preferable, depending on which one would lead to each outcome.

    Eliezer, if a smart creature modifies itself in order to gain strategic advantages from committing itself to future actions, it must think could better achieve its goals by doing so. If so, why should we be concerned, if those goals do not conflict with our goals?

    I think Anonymous, Unknown and Eliezer have been very helpful so far. Following on from them, here is my take:

    There are many ways Omega could be doing the prediction/placement and it may well matter exactly how the problem is set up. For example, you might be deterministic and he is precalculating your choice (much like we might be able to do with an insect or computer program), or he might be using a quantum suicide method, (quantum) randomizing whether the million goes in and then destroying the world iff you pick the wrong option (This will lead to us ... (read more)

    Be careful of this sort of argument, any time you find yourself defining the "winner" as someone other than the agent who is currently smiling from on top of a giant heap.

    This made me laugh. Well said!

    There's only one question about this scenario for me - is it possible for a sufficiently intelligent being to fully, fully model an individual human brain? If so, (and I think it's tough to argue 'no' unless you think there's a serious glass ceiling for intelligence) choose box B. If you try and second-guess (or, hell, googolth-guess) Omega, you're ... (read more)

    How does the box know? I could open B with the intent of opening only B or I could open B with the intent of then opening A. Perhaps Omega has locked the boxes such that they only open when you shout your choice to the sky. That would beat my preferred strategy of opening B before deciding which to choose. I open boxes without choosing to take them all the time.

    Are our common notions about boxes catching us here? In my experience, opening a box rarely makes nearby objects disintegrate. It is physically impossible to "leave $1000 on the table,&qu... (read more)

    Eliezer, if a smart creature modifies itself in order to gain strategic advantages from committing itself to future actions, it must think could better achieve its goals by doing so. If so, why should we be concerned, if those goals do not conflict with our goals?

    Well, there's a number of answers I could give to this:

    *) After you've spent some time working in the framework of a decision theory where dynamic inconsistencies naturally Don't Happen - not because there's an extra clause forbidding them, but because the simple foundations just don't give rise t... (read more)

    So it seems you are disagreeing with most all game theorists in economics as well as most decision theorists in philosophy. Maybe perhaps they are right and you are wrong?

    Maybe perhaps we are right and they are wrong?

    The issue is to be decided, not by referring to perceived status or expertise, but by looking at who has the better arguments. Only when we cannot evaluate the arguments does making an educated guess based on perceived expertise become appropriate.

    Again: how much do we want to bet that Eliezer won't admit that he's wrong in this case? Do we have someone willing to wager another 10 credibility units?

    Caledonian: you can stop talking about wagering credibility units now, we all know you don't have funds for the smallest stake.

    Ben Jones: if we assume that Omega is perfectly simulating the human mind, then when we are choosing between B and A+B, we don't know whether we are in reality or simulation. In reality, our choice does not affect the million, but in the simulation this will. So we should reason "I'd better take only box B, because if this is the simulation then that will change whether or not I get the million in reality".

    There is a big difference between having time inconsistent preferences, and time inconsistent strategies because of the strategic incentives of the game you are playing. Trying to find a set of preferences that avoids all strategic conflicts between your different actions seems a fool's errand.

    What we have here is an inability to recognize that causality no longer flows only from 'past' to 'future'.

    If we're given a box that could contain $1,000 or nothing, we calculate the expected value of the superposition of these two possibilities. We don't actually expect that there's a superposition within the box - we simply adopt a technique to help compensate for what we do not know. From our ignorant perspective, either case could be real, although in actuality either the box has the money or it does not.

    This is similar. The amount of money in the b... (read more)

    How about simply multiplying? Treat Omega as a fair coin toss. 50% of a million is half-a-million, and that's vastly bigger than a thousand. You can ignore the question of whether omega has filled the box, in deciding that the uncertain box is more important. So much more important, that the chance of gaining an extra 1000 isn't worth the bother of trying to beat the puzzle. You just grab the important box.

    After you've spent some time working in the framework of a decision theory where dynamic inconsistencies naturally Don't Happen - not because there's an extra clause forbidding them, but because the simple foundations just don't give rise to them - then an intertemporal preference reversal starts looking like just another preference reversal.

    ... Roughly, self-modifying capability in a classical causal decision theorist doesn't fix the problem that gives rise to the intertemporal preference reversals, it just makes one temporal self win out over all the oth... (read more)

    There is a big difference between having time inconsistent preferences, and time inconsistent strategies because of the strategic incentives of the game you are playing.

    I can see why a human would have time-inconsistent strategies - because of inconsistent preferences between their past and future self, hyperbolic discounting functions, that sort of thing. I am quite at a loss to understand why an agent with a constant, external utility function should experience inconsistent strategies under any circumstance, regardless of strategic incentives. Expected... (read more)

    The entire issue of casual versus inferential decision theory, and of the seemingly magical powers of the chooser in the Newcomb problem, are serious distractions here, as Eliezer has the same issue in an ordinary commitment situation, e.g., punishment. I suggest starting this conversation over from such an ordinary simple example.

    Let me restate: Two boxes appear. If you touch box A, the contents of box B are vaporized. If you attempt to open box B, box A and it's contents are vaporized. Contents as previously specified. We could probably build these now.

    Experimentally, how do we distinguish this from the description in the main thread? Why are we taking Omega seriously when if the discussion dealt with the number of angels dancing on the head of pin the derision would be palpable? The experimental data point to taking box B. Even if Omega is observed delivering the boxes, and making the specified claims regarding their contents, why are these claims taken on faith as being an accurate description of the problem?

    Let's take Bayes seriously.

    Sometime ago there was a posting about something like "If all you knew was that the past 5 mornings the sun rose, what would you assign the probability the that sun would rise next morning? It came out so something like 5/6 or 4/5 or so.

    But of course that's not all we know, and so we'd get different numbers.

    Now what's given here is that Omega has been correct on a hundred occasions so far. If that's all we know, we should estimate the probability of him being right next time at about 99%. So if you're a one-boxer your exp... (read more)

    Eliezer, I have a question about this: "There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever. This is a sufficient condition to imply that my utility function is unbounded."

    I can see that this preference implies an unbounded utility function, given that a longer life has a greater utility. However, simply stated in that way, most people might agree with the preference. But consider this gamble instead:

    A: Live 5... (read more)

    If this was the only chance you ever get to determine your lifespan - then choose B. In the real world, it would probably be a better idea to discard both options and use your natural lifespan to search for alternative paths to immortality.
    I disagree, not surprisingly, since I was the author of the comment to which you are responding. I would choose A, and I think anyone sensible would choose A. There's not much one can say here in the way of argument, but it is obvious to me that choosing B here is following your ideals off a cliff. Especially since I can add a few hundred 9s there, and by your argument you should still choose B.

    they would just insist that there is an important difference between deciding to take only box B at 7:00am vs 7:10am, if Omega chooses at 7:05am

    But that's exactly what strategic inconsistency is about. Even if you had decided to take only box B at 7:00am, by 7:06am a rational agent will just change his mind and choose to take both boxes. Omega knows this, hence it will put nothing into box B. The only way out is if the AI self-commits to take only box B is a way that's verifiable by Omega.

    When the stakes are high enough I one-box, while gritting my teeth. Otherwise, I'm more interested in demonstrating my "rationality" (Eliezer has convinced me to use those quotes).

    Perhaps we could just specify an agent that uses reverse causation in only particular situations, as it seems that humans are capable of doing.

    Paul G, almost certainly, right? Still, as you say, it has little bearing on one's answer to the question.

    In fact, not true, it does. Is there anything to stop myself making a mental pact with all my simulation buddies (and 'myself', whoever he be) to go for Box B?

    In arguing for the single box, Yudkowsky has made an assumption that I disagree with: at the very end, he changes the stakes and declares that your choice should still be the same.

    My way of looking at it is similar to what Hendrik Boom has said. You have a choice between betting on Omega being right and betting on Omega being wrong.

    A = Contents of box A

    B = What may be in box B (if it isn't empty)

    A is yours, in the sense that you can take it and do whatever you want with it. One thing you can do with A is pay it for a chance to win B if Omega is right. Y... (read more)

    IMO there's less to Newcomb's paradox than meets the eye. It's basically "A future-predicting being who controls the set of choices could make rational choices look silly by making sure they had bad outcomes". OK, yes, he could. Surprised?

    What I think makes it seem paradoxical is that the paradox both assures us that Omega controls the outcome perfectly, and cues us that this isn't so ("He's already left" etc). Once you settle what it's really saying either way, the rest follows.

    Yes, this is really an issue of whether your choice causes Omega's action or not. The only way for Omega to be a perfect predictor is for your choice to actually cause Omega's action. (For example, Omega 'sees the future' and acts based on your choice). If your choice causes Omega's action, then choosing B is the rational decision, as it causes the box to have the million.

    If your choice does not cause Omega's action, then choosing both boxes is the winning approach. in this case, Omega is merely giving big awards to some people and small awards to ot... (read more)

    the dominant consensus in modern decision theory is that one should two-box...there's a common attitude that "Verbal arguments for one-boxing are easy to come by, what's hard is developing a good decision theory that one-boxes"

    Those are contrary positions, right?

    Robin Hason:
    Punishment is ordinary, but Newcomb's problem is simple! You can't have both.

    The advantage of an ordinary situation like punishment is that game theorists can't deny the fact on the ground that governments exist, but they can claim it's because we're all irrational, which doesn't leave many directions to go in.

    I agree that "rationality" should be the thing that makes you win but the Newcomb paradox seems kind of contrived.

    If there is a more powerful entity throwing good utilities at normally dumb decisions and bad utilities at normally good decisions then you can make any dumb thing look genius because you are under different rules than the world we live in at present.

    I would ask Alpha for help and do what he tells me to do. Alpha is an AI that is also never wrong when it comes to predicting the future, just like Omega. Alpha would examine omega and ... (read more)

    To me, the decision is very easy. Omega obviously possesses more prescience about my box-taking decision than I do myself. He's been able to guess correct in the past, so I'd see no reason to doubt him with myself. With that in mind, the obvious choice is to take box B.

    If Omega is so nearly always correct, then determinism is shown to exist (at least to some extent). That being the case, causality would be nothing but an illusion. So I'd see no problem with it working in "reverse".

    Fascinating. A few days after I read this, it struck me that a form of Newcomb's Problem actually occurs in real life--voting in a large election. Here's what I mean.

    Say you're sitting at home pondering whether to vote. If you decide to stay home, you benefit by avoiding the minor inconvenience of driving and standing in line. (Like gaining $1000.) If you decide to vote, you'll fail to avoid the inconvenience, meanwhile you know your individual vote almost certainly won't make a statistical difference in getting your candidate elected. (Which would be like... (read more)

    A very good point. I'm the type to stay home from the polls. But I'd also one-box..... hm. I think it may have to do with the very weak correlation between my choice to vote and the choice of those of a similar mind to me to vote as opposed to the very strong correlation between my choice to one-box and Omega's choice to put $1,000,000 in box B.
    Rational agents defect against a bunch of irrational fools who are mostly choosing for signalling purposes and who may well vote for the other guy even if they cooperate.

    "If it ever turns out that Bayes fails - receives systematically lower rewards on some problem, relative to a superior alternative, in virtue of its mere decisions - then Bayes has to go out the window."

    What exactly do you mean by mere decisions? I can construct problems where agents that use few computational resources win. Bayesian agents by your own admission have to use energy to get in mutual information with the environment (a state I am still suspecious of), so they have to use energy, meaning they lose.

    The premise is that a rational agent would start out convinced that this story about the alien that knows in advance what they'll decide appears to be false.

    The Kolomogorov complexity of the story about the alien is very large because we have to hypothesize some mechanism by which it can extrapolate the contents of minds. Even if I saw the alien land a million times and watched the box-picking connect with the box contents as they're supposed to, it is simpler to assume that the boxes are some stage magic trick, or even that they are an exception to the u... (read more)

    It is not possible for an agent to make a rational choice between 1 or 2 boxes if the agent and Omega can both be simulated by Turing machines. Proof: Omega predicts the agent's decision by simulating it. This requires Omega to have greater algorithmic complexity than the agent (including the nonzero complexity of the compiler or interpreter). But a rational choice by the agent requires that it simulate Omega, which requires that the agent have greater algorithmic complexity instead.

    In other words, the agent X, with complexity K(X), must model Omega whi... (read more)

    Um, AIXI is not computable. Relatedly, K(AIXI) is undefined, as AIXI is not a finite object. Also, A can simulate B, even when K(B)>K(A). For example, one could easily define a computer program which, given sufficient computing resources, simulates all Turing machines on all inputs. This must obviously include those with much higher Kolmogorov complexity. Yes, you run into issues of two Turing machines/agents/whatever simulating each other. (You could also get this from the recursion theorem.) What happens then? Simple: neither simulation ever halts.
    Not so. I don't need to simulate a hungry tiger in order to stay safely (and rationally) away from it, even though I don't know the exact methods by which its brain will identify me as a tasty treat. If you think that one can't "rationally" stay away from hungry tigers, then we're using the word "rationally" vastly differently.

    Okay, maybe I am stupid, maybe I am unfamiliar with all the literature on the problem, maybe my English sucks, but I fail to understand the following:
    Is the agent aware of the fact that one boxers get 1 000 000 at the moment Omega "scans" him and presents the boxes?


    Is agent told about this after Omega "has left"?


    Is agent unaware of the fact that Omega rewards one-boxers at all?
    P.S.: Also, as most "decision paradoxes", this one will have different solutions depending on the context (is the agent a starving child in Africa, or a "megacorp" CEO)

    I'm a convinced two-boxer, but I'll try to put my argument without any bias. It seems to me the way this problem has been put has been an attempt to rig it for the one boxers. When we talk about "precommitment" it is suggested the subject has an advance knowledge of Omega and what is to happen. The way I thought the paradox worked, was that Omega would scan/analyze a person and make its prediction, all before the person ever heard of the dilemna. Therefore, a person has no way to develop an intention of being a one-boxer or a two-boxer t... (read more)

    The key point you've missed in your analysis, however, is that Omega is almost always correct in his predictions. It doesn't matter how Omega does it - that is a separate problem. You don't have enough information about his process of prediction to make any rational judgment about it except for the fact that it is a very, very good process. Brain scans, reversed causality, time travel, none of those ideas matter. In the paradox as originally posed, all you have are guesses about how he may have done it, and you would be an utter fool to give higher weight to those guesses than to the fact that Omega is always right. The if observations (that Omega is always right) disagree with theory (that Omega cannot possibly be right), it is the theory that is wrong, every time. Thus the rational agent should, in this situation, give extremely low weight to his understanding of the way the universe works, since it is obviously flawed (the existence of a perfect predictor proves this). The question really comes down to 100% chance of getting $1000 plus a nearly 0% chance of getting $1.01 million, vs nearly 100% chance of getting $1 million. What really blows my mind about making the 2-box choice is that you can significantly reduce Omega's ability to predict the outcome, and unless you are absolutely desperate for that $1000* the 2-box choice doesn't become superior until Omega is only roughly 50% accurate (at 50.1% the outcome equalizes). Only then do you expect to get more money, on average, by choosing both boxes. In other words, if you think Omega is doing anything but flipping a coin to determine the contents of box B, you are better off choosing box B. *I could see the value of $1000 rising significantly if, for example, a man is holding a gun to your head and will kill you in two minutes if you don't give him $1000. In this case, any uncertainty of Omega's abilities are overshadowed by the certainty of the $1000. This inverts if the man with the gun is demanding more

    If the alien is able to predict your decision, it follows that your decision is a function of your state at the time the alien analyzes you. Then, there is no meaningful question of "what should you do?" Either you are in a universe in which you are disposed to choose the one box AND the alien has placed the million dollars, or you are in a universe in which you are disposed to take both boxes AND the alien has placed nothing. If the former, you will have the subjective experience of "deciding to take the one box", which is itself a det... (read more)

    Yes, but when I tried to write it up, I realized that I was starting to write a small book. And it wasn't the most important book I had to write, so I shelved it. My slow writing speed really is the bane of my existence. The theory I worked out seems, to me, to have many nice properties besides being well-suited to Newcomblike problems. It would make a nice PhD thesis, if I could get someone to accept it as my PhD thesis. But that's pretty much what it would take to make me unshelve the project. Otherwise I can't justify the time expenditure, not at ... (read more)

    Isn't this the exact opposite arguement from the one that was made in Dust Specks vs 50 Years of Torture?

    Correct me if I'm wrong, but the argument in this post seems to be "Don't cling to a supposedly-perfect 'causal decision theory' if it would make you lose gracefully, take the action that makes you WIN."

    And the argument for preferring 50 Years of Torture over 3^^^3 Dust Specks is that "The moral theory is perfect. It must be clung to, even when the result is a major loss."

    How can both of these be true?

    (And yes, I am defining "pr... (read more)

    One belated point, some people seem to think that Omega's successful prediction is virtually impossible and that the experiment is a purely fanciful speculation. However it seems to me entirely plausible that having you fill out a questionnaire while being brain scanned might well bring this situation into practicality in the near future. The questions, if filled out correctly, could characterize your personality type with enough accuracy to give a very strong prediction about what you will do. And if you lie, in the future that might be detected with a br... (read more)

    Somehow I'd never thought of this as a rationalist's dilemma, but rather a determinism vs free will illustration. I still see it that way. You cannot both believe you have a choice AND that Omega has perfect prediction.

    The only "rational" (in all senses of the word) response I support is: shut up and multiply. Estimate the chance that he has predicted wrong, and if that gives you +expected value, take both boxes. I phrase this as advice, but in fact I mean it as prediction of rational behavior.

    In my motivations and in my decision theory, dynamic inconsistency is Always Wrong. Among other things, it always implies an agent unstable under reflection.

    If you really want to impress an inspector who can see your internal state, by altering your utility function to conform to their wishes, then one strategy would be to create a trusted external "brain surgeon" agent with the keys to your utility function to change it back again after your utility function has been inspected - and then forget all about the existence of the surgeon.

    The inspector will be able to see the lock on your utility function - but those are pretty standard issue.

    As a rationalist, it might be worthwhile to take the one box just so those Omega know-it-alls will be wrong for once.

    If random number generators not determinable by Omega exist, generate one bit of entropy. If not, take the million bucks. Quantum randomness anyone?

    Given how many times Eliezer has linked to it, it's a little surprising that nobody seems to have picked up on this yet, but the paragraph about the utility function not being up for grabs seems to have a pretty serious technical flaw:

    There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever. This is a sufficient condition to imply that my utility function is unbounded.

    Let p = 80% and let q be one in a million. I'm pretty... (read more)

    Benja, the notion is that "live forever" does not have any finite utility, since it is bounded below by a series of finite lifetimes whose utility increases without bound.

    thinks -- Okay, so if I understand you correctly now, the essential thing I was missing that you meant to imply was that the utility of living forever must necessarily be equal to (cannot be larger than) the limit of the utilities of living a finite number of years. Then, if u(live forever) is finite, p times the difference between u(live forever) and u(live n years) must become arbitrarily small, and thus, eventually smaller than q times the difference between u(live n years) and u(live googolplex years). You then arrive at a contradiction, from which you... (read more)

    There are two ways of thinking about the problem.

    1. You see the problem as decision theorist, and see a conflict between the expected utility recommendation and the dominance principle. People who have seen the problem this way have been led into various forms of causal decision theory.

    2. You see the problem as game theorist, and are trying to figure out the predictor's utility function, what points are focal and why. People who have seen the problem this way have been led into various discussions of tacit coordination.

    Newcomb's scenario is a paradox, ... (read more)

    Re: First, foremost, fundamentally, above all else: Rational agents should WIN.

    When Deep Blue beat Gary Kasparov, did that prove that Gary Kasparov was "irrational"?

    It seems as though it would be unreasonable to expect even highly rational agents to win - if pitted against superior competition. Rational agents can lose in other ways as well - e.g. by not having access to useful information.

    Since there are plenty of ways in which rational agents can lose, "winning" seems unlikely to be part of a reasonable definition of rationality.

    I think I've solved it.

    I'm a little late to this, and given the amount of time people smarter than myself have spent thinking about this it seems naive even to myself to think that I have found a solution to this problem. That being said, try as I might, I can't find a good counter argument to this line of reasoning. Here goes...

    The human brain's function is still mostly a black box to us, but the demonstrated predictive power of this alien is strong evidence that this is not the case with him. If he really can predict human decisions, than the mere fact ... (read more)

    Cross-posting from Less Wrong, I think there's a generalized Russell's Paradox problem with this theory of rationality:

    I don't think I buy this for Newcomb-like problems. Consider Omega who says, "There will be $1M in Box B IFF you are irrational."

    Rationality as winning is probably subject to a whole family of Russell's-Paradox-type problems like that. I suppose I'm not sure there's a better notion of rationality.

    Eliezer, why didn't you answer the question I asked at the beginning of the comment section of this post?

    The 'delayed choice' experiments of Wheeler & others appear to show a causality that goes backward in time. So, I would take just Box B.

    I would use a true quantum random generator. 51% of the time I would take only one box. Otherwise I would take two boxes. Thus Omega has to guess that I will only take one box, but I have a 49% chance of taking home another $1000. My expected winnings will be $1000490 and I am per Eliezer's definition more rational than he.

    This is why I restate the problem to exclude the million when people choose randomly.

    I'm a bit nervous, this is my first comment here, and I feel quite out of my league.

    Regarding the "free will" aspect, can one game the system? My rational choice would be to sit right there, arms crossed, and choose no box. Instead, having thus disproved Omega's infallibility, I'd wait for Omega to come back around, and try to weasel some knowledge out of her.

    Rationally, the intelligence that could model mine and predict my likely action (yet fail to predict my inaction enough to not bother with me in the first place), is an intelligence I'd like... (read more)

    Hi. This is a rather old post, so you might not get too many replies. Newcomb's problem often comes with the caveat that, if Omega thinks you're going to game the system, it will leave you with only the $1,000. But yes, we like clever answers here, although we also like to consider, for the purposes of thought experiments, the least convenient possible world in which the loopholes we find have been closed. Also, may I suggest visiting the welcome thread?

    I've come around to the majority viewpoint on the alien/Omega problem. It seems to be easier to think about when you pin it down a bit more mathematically.

    Let's suppose the alien determines the probability of me one-boxing is p. For the sake of simplicity, let's assume he then puts the 1M into one of the boxes with this probability p. (In theory he could do it whenever p exceeded some thresh-hold, but this just complicates the math.)

    Therefore, once I encounter the situation, there are two possible states:

    a) with probability p there is 1M in one box, and 1k... (read more)

    There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever. This is a sufficient condition to imply that my utility function is unbounded.

    Wait a second, the following bounded utility function can explain the quoted preferences:

    • U(live googolplex years) = 99
    • limit as N goes to infinity of U(live N years) = 100
    • U(live forever) = 101

    Benja Fallenstein gave an alternative formulation that does imply an unbounded utility function:

    For all n, there is an even larger n' such that (p+q)*u(live n years) < p*u(live n' years) + q*(live a googolplex years).

    But these preferences are pretty counter-intuitive to me. If U(live n years) is unbounded, then the above must hold for any nonzero p, q, and with "googolplex" replaced by any finite number. For example, let p = 1/3^^^3, q = .8, n = 3^^^3, and replace "googolplex" with "0". Would you really be willing to give up .8 probability of 3^^^3 years of life for a 1/3^^^3 chance at a longer (but still finite) one? And that's true no matter how many up-arrows we add to these numbers?

    "Would you really be willing to give up .8 probability of 3^^^3 years of life for a 1/3^^^3 chance at a longer (but still finite) one?" I'd like to hear this too.
    7Eliezer Yudkowsky14y
    Okay. There's two intuitive obstacles, my heuristic as a human that my mind is too weak to handle tiny probabilities and that I should try to live my life on the mainline, and the fact that 3^^^3 already extrapolates a mind larger than the sum of every future experience my present self can empathize with. But I strongly suspect that answering "No" would enable someone to demonstrate circular / inconsistent preferences on my part, and so I very strongly suspect that my reflective equilibrium would answer "Yes". Even in the realm of the computable, there are simple computable functions that grow a heck of a lot faster than up-arrow notation.

    Eliezer, would you be willing to bet all of your assets and future earnings against $1 of my money, that we can do an infinite amount of computation before the universe ends or becomes incapable of supporting life?

    Your answer ought to be yes, if your preferences are what you state. If it turns out that we can do an infinite amount of computation before the universe ends, then this bet increases your money by $1, which allows you to increase your chance of having an infinite lifetime by some small but non-zero probability. If it turns out that our universe can't do an infinite amount of computation, you lose a lot, but the loss of expected utility is still tiny compared to what you gain.

    So, is it a bet?

    Also, why do you suspect that answering "No" would enable someone to demonstrate circular / inconsistent preferences on your part?

    2Eliezer Yudkowsky14y
    No for two reasons - first, I don't trust human reason including my own when trying to live one's life inside tiny probabilities of huge payoffs; second, I ordinarily consider myself an average utilitarian and I'm not sure this is how my average utilitarianism plays out. It's one matter if you're working within a single universe in which all-but-infinitesimal of the value is to be found within those lives that are infinite, but I'm not sure I would compare two differently-sized possible Realities the same way. I am not sure I am willing to say that a finite life weighs nothing in my utility function if an infinite life seems possible - though if both were known to coexist in the same universe, I might have to bite that bullet. (At the opposite extreme, a Bostromian parliament might assign both cases representative weight proportional to probability and let them negotiate the wise action.) Also I have severe doubts about infinite ethics, but that's easily fixed using a really large finite number instead (pay everything if time < googolplex, keep $1 if time > TREE(100), return $1 later if time between those two bounds). Keep growing the lifespan by huge computational factors, keep slicing near-infinitesimally tiny increments off the probability. (Is there an analogous inconsistency to which I expose myself by answering "No" to the bet above, from trying to treat alternative universes differently than side-by-side spatial reasons?)
    0Wei Dai14y
    In that case, it's not that your utility function is unbounded in years lived, but rather your utility for each year lived is a decreasing function of the lifetime of the universe (or perhaps total lifetime of everyone in the universe). I'll have to think if that makes sense.
    2Eliezer Yudkowsky14y
    It's possible that I'm reasoning as if my utility function is over "fractions of total achievable value" within any given universe. I am not sure if there are any problems with this, even if it's true.
    0Wei Dai14y
    After thinking about it, that doesn't make sense either. Suppose Omega comes to you and says that among the universes that you live in, there is a small fraction that will end in 5 years. He offers to kill you now in those universes, in exchange for granting you a googleplex years of additional life in a similar fraction of universes with time > TREE(100) and where you would have died in less than googleplex years without his help (and where others manage to live to TREE(100) years old if that makes any difference). Would you refuse?
    2Eliezer Yudkowsky14y
    No. But here, by specification, you're making all the universes real and hence part of a larger Reality, rather than probabilities of which only a single one is real. If there were only one Reality, and there were small probabilities of it being due to end in 5 years, or in a googolplex years, and the two cases seemed of equal probability, and Omega offered to destroy reality now if it were only fated to last 5 years, in exchange for extending its life to TREE(100) if it were otherwise fated to last a googolplex years... well, this Reality is already known to have lasted a few billion years, and through, say, around 2 trillion life-years, so if it is due to last only another 5 years the remaining 30 billion life-years are not such a high fraction of its total value to be lost - we aren't likely to do so much more in just another 5 years, if that's our limit; it seems unlikely that we'd get FAI in that time. I'd probably still take the offer. But I wouldn't leap at it.
    0Wei Dai14y
    In that case, would you accept my original bet if I rephrase it as making all the universes part of a larger Reality? That is, if in the future we have reason to believe that Tegmark's Level 4 Multiverse is true, and find ourselves living in a universe with time < googolplex, then you'd give you all your assets and future earnings, in return for $1 of my money if we find ourselves living in a universe with time > TREE(100).
    1Eliezer Yudkowsky14y
    I wouldn't, but my reflective equilibrium might very well do so. I wouldn't due to willpower failure exceeding benefit of $1 if I believe my mainline probability is doomed to eternal poverty. Reflective equilibrium probably would, presuming there's a substantial probability of >TREE(100), or that as a limiting process the "tiny" probability falls off more slowly than the "long-lived" universe part increases. On pain of inconsistency when you raise the lifespan by large computational factors each time, and slice tiny increments off the probability each time.
    3Wei Dai14y
    Ok, as long as your utility function isn't actually unbounded, here's what I think makes more sense, assuming a Level 4 Multiverse. It's also a kind of "fractions of total achievable value". Each mathematical structure representing a universe has a measure, which represents it's "fraction of all math". (Perhaps it's measure is exponential in zero minus the length of its definition in a formal set theory.) My utility over that structure is bounded by this measure. In other words, if that structure represents my idea of total utopia, when my utility for it would be its measure. If it's total dystopia, my utility for it would be 0. Within a universe, different substructures (for example branches or slices of time) also have different measures, and if I value such substructures independently, my utilities for them are also bounded by their measures. For example, in a universe that ends at t = TREE(100), a time slice with t < googolplex has a much higher measure than a random time slice (since it takes more bits to represent a random t). If I value each person independently (and altruistically), then it's like average utilitarianism, except each person is given a weight equal to its measure instead of 1/population. This proposal has its own counter-intuitive implications, but overall I think it's better than the alternatives. It fits in nicely with MWI. It also manages to avoid running into problems with infinities.
    6Eliezer Yudkowsky14y
    I have to say this strikes me as a really odd proposal, though it's certainly interesting from the perspective of the Doomsday Argument if advanced civilizations have a thermodynamic incentive to wait until nearly the end of the universe before using their hoarded negentropy. But for me it's hard to see why "reality-fluid" (the name I give your "measure", to remind myself that I don't understand it at all) should dovetail so neatly with the information needed to locate events in universes or universes in Level IV. It's clear why an epistemic prior is phrased this way - but why should reality-fluid behave likewise? Shades of either Mind Projection Fallacy or a very strange and very convenient coincidence.

    Actually, I think I can hazard a guess to that one. I think the idea would be "the simpler the mathematical structure, the more often it'd show up as a substructure in other mathematical structures"

    For instance, if you are building large random graphs, you'd expect to see some specific pattern of, say, 7 vertices and 18 edges show up as subgraphs more often then, say, some specific pattern of 100 vertices and 2475 edges.

    There's a sense in which "reality fluid" could be distributed evenly which would lead to this. If every entire mathematical structure got an equal amount of reality stuff, then small structures would benefit from the reality juice granted to the larger structures that they happen to also exist as substructures of.

    EDIT: blargh, corrected big graph edge count. meant to represent half a complete graph.

    3Wei Dai14y
    Well, why would it be easier to locate some events or universes than others, unless they have more reality-fluid? Why is it possible to describe one mathematical structure more concisely than another, or to specify one computation using less bits than another? Is that just a property of the mind that's thinking about these structures and computations, or is it actually a property of Reality? The latter seems more likely to me, given results in algorithmic information theory. (I don't know if similar theorems has been or can be proven about set theory, that the shortest description lengths in different formalizations can't be too far apart, but it seems plausible.) Also, recall that in UDT, there is no epistemic prior. So, the only way to get an effect similar to EDT/CDT w/ universal prior, is with a weighting scheme over universes/events like I described.
    2Eliezer Yudkowsky14y
    I can sort of buy the part where simple universes have more reality-fluid, though frankly the whole setup strikes me as a mysterious answer to a mysterious question. But the part where later events have less reality-fluid within a single universe, just because they take more info to locate - that part in particular seems really suspicious. MPF-ish.
    1Wei Dai14y
    I'm far from satisfied with the answer myself, but it's the best I've got so far. :)
    Consider the case where you are trying to value (a) just yourself versus (b) the set of all future yous that satisfy the constraint of not going into negative utility. The shannon information of the set (b) could be (probably would be) lower than that of (a). To see this, note that the complexity (information) of the set of all future yous is just the info required to specify (you,now) (because to compute the time evolution of the set, you just need the initial condition), whereas the complexity (information) of just you is a series of snapshots (you, now), (you, 1 microsecond from now), ... . This is like the difference between a JPEG and an MPEG. The complexity of the constraint probably won't make up for this. If the constraint of going into negative utility is particularly complex, one could pick a simple subset of nonnegative utility future yous, for example by specifying relatively simple constraints that ensure that the vast majority of yous satisfying those constraints don't go into negative utility. This is problematic because it means that you would assign less value to a large set of happy future yous than to just one future you. A large and exhaustive set of future happy yous is less complex (easier to specify) than just one.
    Related: That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradox (
    That does have quite a bit of intuitive appeal! However, when you look at a possible universe from the outside, there are no levers nor knobs you can turn, and all the value achieved by the time of heat death was already inherent in the configurations right after the big bang-- --so if you do not want "fraction of total achievable value" to be identically one for every possible universe, the definition of your utility function seems to get intertwined with how exactly you divvy up the world into "causal nodes" and "causal arrows", in a way that does not seem to happen if you define it in terms of properties of the outcome, like how many fulfilling lifes lived. (Of course, being more complicated doesn't imply being wrong, but it seems worth noting.) And yes, I'm taking a timeful view for vividness of imagination, but I do not think the argument changes much if you don't do that; the point is that it seems like number-of-fulfilling-lifes utility can be computed given only the universal wavefunction as input, whereas for fraction-of-achievable-fulfilling-lifes, knowing the actual wavefunction isn't enough. Could your proposal lead to conflicts between altruists who have the same values (e.g. number of fulfilling lifes), but different power to influence the world (and thus different total achievable value)?
    This looks pretty plausible to me, because it does seem there is some disutility to the simple fact of dying, regardless of how far in the future that happens. So U(live N years) always contains that disutility, whereas U(live forever) does not.

    I really don't see what the problem is. Clearly, the being has "read your mind" and knows what you will do. If you are of the opinion to take both boxes, he knows that from his mind scan, and you are playing right into his hands.

    Obviously, your decision cannot affect the outcome because it's already been decided what's in the box, but your BRAIN affected what he put in the box.

    It's like me handing you an opaque box and telling you there is $1 million in it if and only if you go and commit murder. Then, you open the box and find it empty. I then o... (read more)

    The question is how to create a formal decision algorithm that will be able to understand the problem and give the right answer (without failing on other such tests). Of course you can solve it correctly if you are not yet poisoned by too much presumptuous philosophy.
    I guess I'm missing something obvious. The problem seems very simple, even for an AI. The way the problem is usually defined (omega really is omniscient, he's not fooling you around, etc.) there are only two solutions: * You take the two boxes, and Omega had already predicted that, meaning that there is nothing in Box B - you win 1000$ * You take box B only, and Omega had already predicted that, meaning that there is 1M$ in box B - you win 1M$. That's it. Period. Nothing else. Nada. Rien. Nichts. Sod all. These are the only two possible options (again, assuming the hypotheses are true). The decision to take box B only is a simple outcome comparison. It is a perfectly rational decision (if you accept the premises of the game). Now the way Eliezer states it is different from the usual formulation. In Eliezer's version, you cannot be sure about Omega's absolute accuracy. All you know is his previous record. That does complicate things, if only because you might be the victim of a scam (e.g. like the well-known trick to convince comeone that you can consistently predict the winning horse in a 2-horse race - simply start with 2^N people, always give a different prediction to each half of them, discard those to whom you gave the wrong one, etc.) At any rate, the other two outcomes that were impossible in the previous version (involving mis-prediction by Omega) are now possible, with a certain probability that you need to somehow ascertain. That may be difficult, but I don't see any logical paradox. For example, if this happened in the real world, you might reason that the probability that you are being scammed is overwhelming in regard to the probability of existence of a truly omniscient predictor. This is a reasonable inference from the fact that we hear about scams every day, but nobody has ever reported such an omniscient predictor. So you would take both boxes and enjoy your expected $1000+epsilon (Omega may have been sincere but deluded, lucky in the

    I one-box, but not because I haven't considered the two-box issue.

    I one-box because it's a win-win in the larger context. Either I walk off with a million dollars, OR I become the first person to outthink Omega and provide new data to those who are following Omega's exploits.

    Even without thinking outside the problem, Omega is a game-breaker. We do not, in the problem as stated, have any information on Omega other than that they are superintelligent and may be able to act outside of casuality. Or else Omega is simply a superduperpredictor, to the point wher... (read more)

    My solution to the problem of the two boxes:

    Flip a coin. If heads, both A & B. If tails, only A. (If the superintelligence can predict a coin flip, make it a radioactive decay or something. Eat quantum, Hal.)

    In all seriousness, this is a very odd problem (I love it!). Of course two boxes is the rational solution - it's not as if post-facto cogitation is going to change anything. But the problem statement seems to imply that it is actually impossible for me to choose the choice I don't choose, i.e., choice is actually impossible.

    Something is absurd here. I suspect it's the idea that my choice is totally predictable. There can be a random element to my choice if I so choose, which kills Omega's plan.

    It is a common assumption in these sorts of problems that if Omega predicts that you will condition your choice on a quantum event, it will not put the money in Box B. See The Least Convenient Possible World.
    At face, that does sound absurd. The problem is that you are underestimating a superintelligence. Imagine that the universe is a computer simulation, so that a set of physical laws plus a very, very long string of random numbers is a complete causal model of reality. The superintelligence knows the laws and all of the random numbers. You still make a choice, even though that choice ultimately depends on everything that preceded it. See I think much of the debate about Newcomb's Problem is about the definition of superintelligence.
    No it isn't. If you like money it is rational to get more money. Take one box.
    What wedrifid said. See also Rationality is Systematized Winning and the section of What Do We Mean By "Rationality"? about "Instrumental Rationality", which is generally what we mean here when we talk about actions being rational or irrational. If you want to get more money, than the instrumentally rational action is the epistemically rational answer to the question "What course of action will cause me to get the most money?". If you accept the premises of Omega thought experiments, then the right answer is one-boxing, period. If you don't accept the premises, it doesn't make sense for you to be answering it one way or the other.
    I thought about this last night and also came to the conclusion that randomizing my choice would not "assume the worst" as I ought to. And I fully accept that this is just a thought experiment & physics is a cheap way out. I will now take the premises or leave them. :)

    I'm not reading 127 comments, but as a newcomer who's been invited to read this page, along with barely a dozen others, as an introduction, I don't want to leave this unanswered, even though what I have to say has probably already been said.

    First of all, the answer to Newcomb's Problem depends a lot on precisely what the problem is. I have seen versions that posit time travel, and therefore backwards causality. In that case, it's quite reasonable to take only one box, because your decision to do so does have a causal effect on the amount in Box B. Presu... (read more)

    You are disposed to take two boxes. Omega can tell. (Perhaps by reading your comment. Heck, I can tell by reading your comment, and I'm not even a superintelligence.) Omega will therefore not put a million dollars in Box B if it sets you a Newcomb's problem, because its decision to do so depends on whether you are disposed to take both boxes or not, and you are.

    I am disposed to take one box. Omega can tell. (Perhaps by reading this comment. I bet you can tell by reading my comment, and I also bet that you're not a superintelligence.) Omega will therefore put a million dollars in Box B if it sets me a Newcomb's problem, because its decision to do so depends on whether I am disposed to take both boxes or not, and I'm not.

    If we both get pairs of boxes to choose from, I will get a million dollars. You will get a thousand dollars. I will be monetarily better off than you.

    But wait! You can fix this. All you have to do is be disposed to take just Box B. You can do this right now; there's no reason to wait until Omega turns up. Omega does not care why you are so disposed, only that you are so disposed. You can mutter to yourself all you like about how silly the problem is; as long as you wander off with just B under your arm, it will tend to be the case that you end the day a millionaire.

    Sometime ago I figured out a refutation of this kind of reasoning in Counterfactual Mugging, and it seems to apply in Newcomb's Problem too. It goes as follows: Imagine another god, Upsilon, that offers you a similar two-box setup - except to get the $2M in the box B, you must be a one-boxer with regard to Upsilon and a two-boxer with regard to Omega. (Upsilon predicts your counterfactual behavior if you'd met Omega instead.) Now you must choose your dispositions wisely because you can't win money from both gods. The right disposition depends on your priors for encountering Omega or Upsilon, which is a "bead jar guess" because both gods are very improbable. In other words, to win in such problems, you can't just look at each problem individually as it arises - you need to have the correct prior/predisposition over all possible predictors of your actions, before you actually meet any of them. Obtaining such a prior is difficult, so I don't really know what I'm predisposed to do in Newcomb's Problem if I'm faced with it someday.
    Something seems off about this, but I'm not sure what.
    I'm pretty sure the logic is correct. I do make silly math mistakes sometimes, but I've tested this one on Vladimir Nesov and he agrees. No comment from Eliezer yet (this scenario was first posted to decision-theory-workshop).
    It reminds me vaguely of Pascal's Wager, but my cached responses thereunto are not translating informatively.
    Then I think the original Newcomb's Problem should remind you of Pascal's Wager just as much, and my scenario should be analogous to the refutation thereof. (Thereunto? :-)
    This is not a refutation, because what you describe is not about the thought experiment. In the thought experiment, there are no Upsilons, and so nothing to worry about. It is if you face this scenario in real life, where you can't be given guarantees about the absence of Upsilons, that your reasoning becomes valid. But it doesn't refute the reasoning about the thought experiment where it's postulated that there are no Upsilons. (Original thread, my discussion.)
    Thanks for dropping the links here. FWIW, I agree with your objection. But at the very least, the people claiming they're "one-boxers" should also make the distinction you make. Also, user Nisan tried to argue that various Upsilons and other fauna must balance themselves out if we use the universal prior. We eventually took this argument to email, but failed to move each other's positions.
    Just didn't want you confusing people or misrepresenting my opinion, so made everything clear. :-)
    OK. I assume the usual (Omega and Upsilon are both reliable and sincere, I can reliably distinguish one from the other, etc.) Then I can't see how the game doesn't reduce to standard Newcomb, modulo a simple probability calculation, mostly based on "when I encounter one of them, what's my probability of meeting the other during my lifetime?" (plus various "actuarial" calculations). If I have no information about the probability of encountering either, then my decision may be incorrect - but there's nothing paradoxical or surprising about this, it's just a normal, "boring" example of an incomplete information problem. I can't see why that is - again, assuming that the full problem is explained to you on encountering either Upsilon or Omega, both are truhful, etc. Why can I not perform the appropriate calculations and make an expectation-maximising decision even after Upsilon-Omega has left? Surely Omega-Upsilon can predict that I'm going to do just that and act accordingly, right?
    Yes, this is a standard incomplete information problem. Yes, you can do the calculations at any convenient time, not necessarily before meeting Omega. (These calculations can't use the information that Omega exists, though.) No, it isn't quite as simple as you state: when you meet Omega, you have to calculate the counterfactual probability of you having met Upsilon instead, and so on.

    Omega lets me decide to take only one box after meeting Omega, when I have already updated on the fact that Omega exists, and so I have much better knowledge about which sort of god I'm likely to encounter. Upsilon treats me on the basis of a guess I would subjunctively make without knowledge of Upsilon. It is therefore not surprising that I tend to do much better with Omega than with Upsilon, because the relevant choices being made by me are being made with much better knowledge. To put it another way, when Omega offers me a Newcomb's Problem, I will condition my choice on the known existence of Omega, and all the Upsilon-like gods will tend to cancel out into Pascal's Wagers. If I run into an Upsilon-like god, then, I am not overly worried about my poor performance - it's like running into the Christian God, you're screwed, but so what, you won't actually run into one. Even the best rational agents cannot perform well on this sort of subjunctive hypothesis without much better knowledge while making the relevant choices than you are offering them. For every rational agent who performs well with respect to Upsilon there is one who performs poorly with respect to anti-Upsilon.... (read more)

    Pascal's Wagers, huh. So your decision theory requires a specific prior?
    In what sense can you update? Updating is about following a plan, not about deciding on a plan. You already know that it's possible to observe anything, you don't learn anything new about environment by observing any given thing. There could be a deep connection between updating and logical uncertainty that makes it a good plan to update, but it's not obvious what it is.
    Huh? Updating is just about updating your map. (?) The next sentence I didn't understand the reasoning of, could you expand?
    Intuitively, the notion of updating a map of fixed reality makes sense, but in the context of decision-making, formalization in full generality proves elusive, even unnecessary, so far. By making a choice, you control the truth value of certain statements—statements about your decision-making algorithm and about mathematical objects depending on your algorithm. Only some of these mathematical objects are part of the "real world". Observations affect what choices you make ("updating is about following a plan"), but you must have decided beforehand what consequences you want to establish ("[updating is] not about deciding on a plan"). You could have decided beforehand to care only about mathematical structures that are "real", but what characterizes those structures apart from the fact that you care about them? Vladimir talks more about his crazy idea in this comment.
    No, that's not what I should do. What I should do is make Omega think that I am disposed to take just Box B. If I can successfully make Omega think that I'll take only Box B but still take both boxes, then I should. But since Omega is superintelligent, let's take it as understood that the only way to make Omega think that I'll take only Box B is to make it so that I'll actually take Box B. Then that is what I should do. But I have to do it now! (I don't do it now only because I don't believe that this situation will ever happen.) Once Omega has placed the boxes and left, if the known laws of physics apply, then it's too late! If you take only Box B and get a million dollars, wouldn't you regret having not also taken Box A? Not only would you have gotten a thousand dollars more, you'd also have shown up that know-it-all superintelligent intergalactic traveller too! That's a chance that I'll never have, since Omega will read my comment here and leave my Box B empty, but you might have that chance, and if so then I hope you'll take it.
    It's not really too late then. Omega can predict what you'll do between seeing the boxes, and choosing which to take. If this is going to include a decision to take one box, then Omega will put a million dollars in that box. I will not regret taking only one box. It strikes me as inconsistent to regret acting as the person I most wish to be, and it seems clear that the person I most wish to be will take only one box; there is no room for approved regret.
    If you say this, then you believe in backwards causality (or a breakdown of the very notion of causality, as in Kevin's comment below). I agree that if causality doesn't work, then I should take only Box B, but nothing in the problem as I understand it from the original post implies any violation of the known laws of physics. If known physics applies, then Omega can predict all it likes, but my actions after it has placed the boxes cannot affect that prediction. There is always the chance that it predicts that I will take both boxes but I take only Box B. There is even the chance that it will predict that I will take only Box B but I take both boxes. Nothing in the problem statement rules that out. It would be different if that were actually impossible for some reason. I knew that you wouldn't, of course, since you're a one-boxer. And we two-boxers will not regret taking both boxes, even if we find Box B empty. Better $1000 than nothing, we will think!
    Beware hidden inferences. Taboo causality.
    I don't see what that link has to do with anything in my comment thread. (I haven't read most of the other threads in reply to this post.) I should explain what I mean by ‘causality’. I do not mean some metaphysical necessity, whereby every event (called an ‘effect’) is determined (or at least influenced in some asymmetric way) by other events (called its ‘causes’), which must be (or at least so far seem to be) prior to the effect in time, leading to infinite regress (apparently back to the Big Bang, which is somehow an exception). I do not mean anything that Aristotle knew enough physics to understand in any but the vaguest way. I mean the flow of macroscopic entropy in a physical system. The best reference that I know on the arrow of time is Huw Price's 1996 book Time's Arrow and Archimedes' Point. But actually I didn't understand how entropy flow leads to a physical concept of causality until several years after I read that, so that might not actually help, and I'm having no luck finding the Internet conversation that made it click for me. But basically, I'm saying that, if known physics applies, then P(there is money in Box B|all information available on a macroscopic level when Omega placed the boxes) = P(there is money in Box B|all information … placed the boxes & I pick both boxes), even though P(I pick both boxes|all information … placed the boxes) < 1, because macroscopic entropy strictly increases between the placing of the boxes and the time that I finally pick a box. So I need to be given evidence that known physics does not apply before I pick only Box B, and a successful record of predictions by Omega will not do that for me.
    Ah, I see what the probem is. You have a confused notion of free will and what it means to make a choice. Making a choice between two options doesn't mean there is a real chance that you might take either option (there always is at least an infinitesimal chance, but that it always true even for things that are not usefully described as a choice). It just means that attributing the reason for your taking whatever option you take is most usefully attributed to you (and not e.g. gravity, government, the person holding a gun to you head etc.). In the end, though, it is (unless the choice is so close that random noise makes the difference) a fact about you that you will make the choice you will make. And it is in principle possible for others to discover this fact about you. If it is a fact about you that you will one-box it is not possible that you will two-box. If it is a fact about you that you will two-box it is not possible that you will one-box. If it is a fact about you that you will leave the choice up to chance then Omega probably doesn't offer you to take part in the first place. Now, when deciding what choice to make it is usually most useful to pretend there is a real possibility of taking either option, since that generally causes facts about you that are more benefitial to you. And that you do that is just another fact about you, and influences the fact about which choice you make. Usually the fact which choice you will make has no consequences before you make your choice, and so you can model the rest of the world as being the same in either case up to that point when counterfactually considering the consequences of either choice. But the fact about which choice you will make is just another fact like any other, and is allowed, even if it usually doesn't, to have consequences before that point in time. If it does it is best, for the very same reason you pretend that either choice is a real possibility in the first place, to also model the rest of the wo
    Alicorn: TobyBartels: I remember reading an article about someone who sincerely lacked respect for people who were 'soft' (not exact quote) on the death penalty ... before ending up on the jury of a death penalty case, and ultimately supporting life in prison instead. It is not inconceivable that a sufficiently canny analyst (e.g. Omega) could deduce that the process of being picked would motivate you to reconsider your stance. (Or, perhaps more likely, motivate a professed one-boxer like me to reconsider mine.)
    From Andy Egan. I would suggest looking at your implicit choice of counterfactuals and their role in your decision theory. Standard causal decision theory involves local violations of the laws of physics (you assign probabilities to the world being such that you'll one-box, or such that you'll one-box, and then ask what miracle magically altering your decision, without any connection to your psychological dispositions, etc, would deliver the highest utility). Standard causal decision theory is a normative principle for action, that says to do the action that would deliver the most utility if a certain kind of miracle happened. But you can get different versions of causal decision theory by substituting different sorts of miracles, e.g. you can say: "if I one-box, then I have a psychology that one-boxes, and likewise for two-boxing" so you select the action such that a miracle giving you the disposition to do so earlier on would have been better. Yet another sort of counterfactual that can be hooked up to the causal decision theory framework would go "there's some mathematical fact about what decision(decisions given Everett) my brain structure leads to in standard physics, and the predictor has access to this mathematical info, so I'll select the action that would be best brought about by a miracle changing that mathematical fact".
    You underestimate the meaning of superintelligence. One way of defining a superintelligence that wins at Newcomb without violating causality, is to assume that the universe is computer simulation like, such that it can be defined by a set of physical laws and a very long string of random numbers. If Omega knows the laws and random numbers that define the universe, shouldn't Omega be able to predict your actions with 100% accuracy? And then wouldn't you want to choose the action that results in you winning a lot more money?
    So part of the definition of a superintelligence is that the universe is like that and Omega knows all that? In other words, if I have convincing evidence that Omega is superintelligent, then I must have convincing evidence that the universe is a computer simulation, etc? Then that changes things; just as the Second Law of Thermodynamics doesn't apply to Maxwell's Demon, so the law of forward causality (which is actually a consequence of the Second Law, under the assumption of no time travel) doesn't apply to a superintelligence. So yes, then I would pick only Box B. This just goes to show how important it is to understand exactly what the problem states.
    The computer simulation assumption isn't necessary, the only thing that matters is that Omega is transcendentally intelligent, and it has all the technology that you might imagine a post-Singularity intelligence might have (we're talking Shock Level 4). So Omega scans your brain by using some technology that is effectively indistinguishable from magic, and we're left to assume that it can predict, to a very high degree of accuracy, whether you're the type of person who would take one or two boxes. Omega doesn't have to actually simulate your underlying physics, it just needs a highly accurate model, which seems reasonably easy to achieve for a superintelligence.
    If its model is good enough that it violates the Second Law as we understand it, fine, I'll pick only Box B, but I don't see anything in the problem statement that implies this. The only evidence that I'm given is that it's made a run of perfect predictions (of unknown length!), is smarter than us, and is from very far away. That's not enough for new physics. And just having a really good simulation of my brain, of the sort that we could imagine doing using known physics but just don't have the technical capacity for, is definitely not good enough. That makes the probability that I'll act as predicted very high, but I'll still come out worse if, after the boxes have been set, I'm unlucky enough to only pick Box B anyway (or come out better if I'm lucky enough to pick both boxes anyway, if Omega pegs me for a one-boxer).

    If its model is good enough that it violates the Second Law as we understand it [...]

    It doesn't have to be even remotely close to good enough to that for the scenario. I'd bet a sufficiently good human psychologist could take omega's role and get it 90%+ right if he tests and interviews the people extensively first (without them knowing the purpose) and gets to exclude people he is unsure about. A super intelligent being should be far, far better at this.

    You yourself claim to know what you would do in the boxing experiment, and you are an agent limited by conventional physics. There is no physical law that forbids another agent from knowing you as well as (or even better than) you know yourself.

    You'll have to explain why you think 99.99% (or whatever) is not good enough, a 0.01% chance to win $ 1000 shouldn't make up for a 99.99% chance of losing $999,000.

    Thanks for the replies, everybody! This is a global response to several replies within my little thread here, so I've put it at nearly the top level. Hopefully that works out OK. I'm glad that FAWS brought up the probabilistic version. That's because the greater the probability that Omega makes mistakes, the more inclined I am to take two boxes. I once read the claim that 70% of people, when told Newcomb's Paradox in an experiment, claim to choose to take only one box. If this is accurate, then Omega can achieve a 70% level of accuracy by predicting that everybody is a one-boxer. Even if 70% is not accurate, you can still make the paradox work by adjusting the dollar amounts, as long as the bias is great enough that Omega can be confident that it will show up at all in the records of its past predictions. (To be fair, the proportion of two-boxers will probably rise as Omega's accuracy falls, and changing the stakes should also affect people's choices; there may not be a fixed point, although I expect that there is.) If, in addition to the problem as stated (but with only 70% probability of success), I know that Omega always predicts one-boxing, then (hopefully) everybody agrees that I should take both boxes. There needs to some correlation between Omega's predictions and the actual outcomes, not just a high proportion of past successes. FAWS also writes: Actually, I don't really want to make that claim. Although I've written things like ‘I would take both boxes’, I really should have written ‘I should take both boxes’. I'm stating a correct decision, not making a prediction about my actual actions. Right now, I predict about a 70% chance of two-boxing given the situation as stated in the original post, although I've never tried to calculate my estimates of probabilities, so who knows what that really means. (H'm, 70% again? Nope, I don't trust that calibration at all!) FAWS writes elsewhere: I don't see what the gun has to do with it; this is a perfectly good

    If Omega is fallible, then the value of one-boxing falls drastically, and even adjusting the amount of money doesn't help in the end;

    Assume Omega has a probability X of correctly predicting your decision:

    If you choose to two-box:

    • X chance of getting $1000
    • (1-X) chance of getting $1,001,000

    If you choose to take box B only:

    • X chance of getting $1,000,000
    • (1-X) chance of getting $0

    Your expected utilities for two-boxing and one-boxing are (respectively):

    E2 = 1000X + (1-X)1001000
    E1 = 1000000X

    For E2 > E1, we must have 1000X + 1,001,000 - 1,001,000X - 1,000,000X > 0, or 1,001,000 > 2,000,000X, or

    X < 0.5005

    So as long as Omega can maintain a greater than 50% accuracy, you should expect to earn more money by one-boxing. Since the solution seems so simple, and since I'm a total novice at decision theory, it's possible I'm missing something here, so please let me know.

    Wait - we can't assume that the probability of being correct is the same for two-boxing and one-boxing. Suppose Omega has a probability X of predicting one when you choose one and Y of predicting one when you choose two. E1 = E($1 000 000) * X E2 = E($1 000) + E($1 000 000) * Y The special case you list corresponds to Y = 1 - X, but in the general case, we can derive that E1 > E2 implies X > Y + E($1 000) / E($1 000 000) If we assume linear utility in wealth, this corresponds to a difference of 0.001. If, alternately, we choose a median net wealth of $93 100 (the U.S. figure) and use log-wealth as the measure of utility, the required difference increases to 0.004 or so. Either way, unless you're dead broke (e.g. net wealth $1), you had better be extremely confident that you can fool the interrogator before you two-box.
    Your caclulation is fine. What you're missing is that Omega has a record of 70% accuracy because Omega always predicts that a person will one-box and 70% of people are one-boxers. So Omega always puts the million dollars in Box B, and I will always get $1,001,000$ if I'm one of the 30% of people who two-box. At least, that is a possibility, which your calculation doesn't take into account. I need evidence of a correlation between Omega's predictions and the participants' actual behaviour, not just evidence of correct predictions. My prior probability distribution for how often people one-box isn't even concentrated very tightly around 70% (which is just a number that I remember reading once as the result of one survey), so anything short of a long run of predictions with very high proportion of correct ones will make me suspect that Omega is pulling a trick like this. So the problem is much cleaner as Eliezer states it, with a perfect record. (But if even that record is short, I won't buy it.)
    Oops, I see that RobinZ already replied, and with calculations. This shows that I should still remove the word ‘drastically’ from the bit that nhamann quoted.

    There is a good chance I am missing something here, but from an economic perspective this seems trivial:

    P(Om) is the probability the person assigns Omega of being able to accurately predict their decision ahead of time.

    A. P(Om) x $1m is the expected return from opening one box.

    B. (1 - P(Om))x$1m + $1000 is the expected return of opening both boxes (the probability that Omega was wrong times the million plus the thousand.)

    Since P(Om) is dependent on people's individual belief about Omega's ability to predict their actions it is not surprising different peop... (read more)

    Re: "Do you take both boxes, or only box B?"

    It would sure be nice to get hold of some more data about the "100 observed occasions so far". If Omega only visits two-boxers - or tries to minimise his outgoings - it would be good to know that. Such information might well be accessible - if we have enough information about Omega to be convinced of his existence in the first place.

    What this is really saying is “if something impossible (according to your current theory of the world) actually happens, then rather than insisting it’s impossible and ignoring it, you should revise your theory to say that’s possible”. In this case, the impossible thing is reverse causality; since we are told of evidence that reverse causality has happened in the form of 100 successful previous experiments, we must revise our theory to accept that reverse causality actually can happen. This would lead us to the conclusion that we should take one box. Alter... (read more)

    The link to that thesis doesn't seem to work for me.

    A quick google turned up one that does

    You know, I honestly don't even understand why this is a point of debate. One boxing and taking box B (and being the kind of person who will predictably do that) seem so obviously like the rational strategy that it shouldn't even require explanation.

    And not obvious in the same way most people think the monty hill problem (game show, three doors, goats behind two, sports-car behind one, ya know?) seems 'obvious' at first.

    In the case of the monty hill problem, you play with it, and the cracks start to show up, and you dig down to the surprising truth.

    In this case, I don't see how anyone could see and cracks in the first place.

    Am I missing something here?

    It is the obvious rational strategy... which is why using a decision theory that doesn't get this wrong is important.
    Yup yup, you're right, of course. What I was trying to say, then, is that I don't understand why there's any debate about the validity of a decision theory that gets this wrong. I'm surprised everyone doesn't just go, "Oh, obviously any decision theory that says two-boxing is 'rational' is an invalid theory." I'm surprised that this is a point of debate. I'm surprised, so I'm wondering, what am I missing? Did I manage to make my question clearer like that?
    I can say that for me personally, the hard part - that I did not get past till reading about it here - was noticing that there is actually such a variable as "what decision theory to use"; using a naive CDT sort of thing simply seemed rational /a priori/. Insufficient grasp of the nameless virtue, you could say.
    Meaning you're in the same boat as me? Confused as to why this ever became a point of debate in the first place?
    0Sniffnoy13y I didn't realize that the decision theory could be varied, that the obvious decision theory could be invalid, so I hit a point of confusion with little idea what to do about it.
    But you're not saying that you would ever have actually decided to two-box rather than take box B if you found yourself in that situation, are you? I mean, you would always have decided, if you found yourself in that situation, that you were the kind of person Omega would have predicted to choose box B, right? I am still so majorly confused here. :P
    I have no idea! IIRC I leaned towards one-boxing, but I was honestly confused about it.
    Ahah. So do you remember if you were confused in yourself, for reasons generated by your own brain, or just by your knowledge that some experts were saying two-boxing was the 'rational' strategy?
    It's a good question. You aren't missing anything. And "people are crazy, the world is mad" isn't always sufficient. ;)
    Ha! =] Okay, I DO expect to see lots of 'people are crazy, the world is mad' stuff, yeah, I just wouldn't expect to see it on something like this from the kind of people who work on things like Causal Decision Theory! :P So I guess what I really want to do first is CHECK which option is really most popular among such people: two-boxing, or predictably choosing box B? Problem is, I'm not sure how to perform that check. Can anyone help me there?
    It is fairly hard to perform such checks. We don't have many situations which are analogous to Newcomb's problem. We don't have perfect predictors and most situations humans are in can be considered "iterated". At least, we can consider most people to be using their 'iterated' reasoning by mistake when we put them in once off situations. The closest analogy that we can get reliable answers out of is the 'ultimatum game' with high stakes... in which people really do refuse weeks worth of wages. By the way, have you considered what you would do if the boxes were transparent? Just sitting there. Omega long gone and you can see piles of cash in front of you... It's tricky. :)
    Thanks, but I meant not a check on what these CDT-studying-type people would DO if actually in that situation, but a check on whether they actually say that two-boxing would be the "rational" thing to do in that hypothetical situation. I haven't considered you transparency question, no. Does that mean Omega did exactly what he would have done if the boxes were opaque, except that they are in fact transparent (a fact that did not figure into the prediction)? Because in that case I'd just see the million in B, and the thousand in A, and of course take 'em both. Otherwise, Omega should be able to predict as well as me that, if I knew the rules of this game were that, if I decided to predictably choose to take only box B and leave A alone, box B would contain a million, and both boxes are transparent (and this transparency is figured into the prediction), I would expect to see a million in box B, take it, and just walk away from the paltry thousand in A. This make sense?
    Suppose my decision algorithm for the "both boxes are transparent" case is to take only box B if and only if it is empty, and to take both boxes if and only if box B has a million dollars in it. How does Omega respond? No matter how it handles box B, it's implied prediction will be wrong. Perhaps just as slippery, what if my algorithm is to take only box B if and only if it contains a million dollars, and to take both boxes if and only if box B is empty? In this case, anything Omega predicts will be accurate, so what prediction does it make? Come to think of it, I could implement the second algorithm (and maybe the first) if a million dollars weighs enough compared to the boxes. Suppose my decision algorithm outputs: "Grab box B and test it's weight, and maybe shake it a bit. If it clearly has a million dollars in it, take only box B. Otherwise, take both boxes." If that's my algorithm, then I don't think the problem actually tells us what Omega predicts, and thus what outcome I'm getting.
    In the first case, Omega does not offer you the deal, and you receive $0, proving that it is possible to do worse than a two-boxer. In the second case, you are placed into a superposition of taking one box and both boxes, receiving the appropriate reward in each. In the third case, you are counted as 'selecting' both boxes, since it's hard to convince Omega that grabbing a box doesn't count as selecting it.
    The premise is that Omega offers you the deal. If Omega's predictions are always successful because it won't offer the deal when it can't predict the result, you can use me as Omega and I'd do as well as him--I just never offer the deal. The (non-nitpicked version of the) transparent box case shows what's wrong with the concept: Since your strategy might involve figuring out what Omega would have done, it may be in principle impossible for Omega to predict what you're going to do, as Omega is indirectly trying to predict itself, leading to an undecideability paradox. The transparent boxes just make this simpler because you can "figure out" what Omega would have done by looking into the transparent boxes. Of course, if you are not a perfect reasoner, it might be possible that Omega can always predict you, but then the question is no longer "which choice should I make", it's "which choice should I make within the limits of my imperfect reasoning". And answering that requires formalizing exactly how your reasoning is limited, which is rather hard.
    The naive presentation of the transparent problem is circular, and for that reason ill defined (what you do depends on what's in the boxes depends on omega's prediction depends on what you do...). A plausible version of the transparent newcomb's problem involves Omega: 1. Predicting what you'd do if you saw box B full (and never mind the case where box B is empty). 2. Predicting what you'd do if you saw box B empty (and never mind the case where box B is full). 3. Predicting what you'd do in both cases, and filling box B if and only if you'd one-box in both of them. Or variations of those. There's no circularity when he only makes such "conditional" predictions. He could use the same algorithms in the non-transparent case, and they would reduce to the normal newcomb's problem usually, but prevent you from doing any tricky business if you happen to bring an X-ray imager (or kitchen scales) and try to observe the state of box B.
    Death by lightning. I typically include such disclaimers such as the above in a footnote or more precisely targeted problem specification so as to avoid any avoid-the-question technicalities. The premise is not that Omega is an idiot or a sloppy game-designer. You took box B. Putting it down again doesn't help you. Finding ways to be cleverer than Omega is not a winning solution to Newcomblike problems.
    * Box B appears full of money; however, after you take both boxes, you find that the money in Box B is Monopoly money. The money in Box A remains genuine, however. * Box B appears empty, however, on opening it you find, written on the bottom of the box, the full details of a bank account opened by Omega, containing one million dollars, together with written permission for you to access said account. In short, even with transparent boxes, there's a number of ways for Omega to lie to you about the contents of Box B, and in this manner control your choice. If Omega is constrained to not lie about the contents of Box B, then it gets a bit trickier; Omega can still maintain an over 90% success rate by presenting the same choice to plenty of other people with an empty box B (since most people will likely take both boxes if they know B is empty). Or, alternatively, Omega can decide to offer you the choice at a time when Omega predicts you won't live long enough to make it. That depends; instead of making a prediction here, Omega is controlling your choice. Whether you get the million dollars or not in this case depends on whether Omega wants you to have the million dollars or not, in furtherance of whatever other plans Omega is planning. Omega doesn't need to predict your choice; in the transparent-box case, Omega needs to predict your decision algorithm.
    "The boxes are transparent" doesn't literally mean "light waves pass through the boxes" given the description of the problem; it means "you can determine what's inside the boxes without (and before) opening them". Responding by saying "maybe you can see into the boxes but you can't tell if the money inside is fake" is being hyper-literal and ignoring what people really mean when they specify "suppose the boxes are transparent".
    Fair enough. I am at times overly literal. In which case, if you are determined to show that Omega's prediction is incorrect, and Omega can predict that determination, then the only way that Omega can avoid making an incorrect prediction is either to modify you in some manner (until you are no longer determined to make Omega's prediction incorrect), or to deny you the chance to make the choice entirely. For example, Omega might modify you by changing your circumstances; e.g. giving a deadly disease to someone close to you; which can be cured, but only at a total cost of all the money you are able to raise plus $1000. If Omega then offers the choice (with box B empty) most people would take both boxes, in order to be able to afford the cure. Alternatively, given such a contrary precommitment, Omega may simply never offer you the choice at all; or might offer you the choice three seconds before you get struck by lightning.
    "Omega puts money inside the boxes, you just never live to get it" is as outside the original problem as "the boxes are transparent, you just don't understand what you're seeing when you look in them" is outside the transparent problem. Just because the premise of the problem doesn't explicitly say "... and you get the contents of the boxes" doesn't mean the paradox can be resolved by saying you don't get the contents of the boxes--that's being hyper-literal again. Likewise, just because the problem doesn't say "... and Omega can't modify you to change your choice" doesn't mean that the paradox can be resolved by saying that Omega can modify change your choice--the problem is about decision theory, and Omega doesn't have capabilities that are irrelevant to what the problem is about.
    The problem, as stated, as far as I can tell gives Omega three options: * Fail to correctly predict what the person will choose * Refuse to participate * Cheat It is likely that Omega will try to correctly predict what the person will choose; that is, Omega will strive to ignore the first option. If Omega offers the choice to this hypothetical person in the first place, then Omega is not taking the second option. That leaves the third option; to cheat. I expect that this is the choice that Omega will be most likely to take; one of the easiest ways to do this is by ignoring the spirit of the constraints and taking the exact literal meaning. (Another way is to creatively misunderstand the spirit of the rules as given). So I provided some suggestions with regard to how Omega might cheat; such as arranging that the decision is never made. If you think that's outside the problem, then I'm curious; what do you think Omega would do?
    The constraints aren't constraints on Omega; the constraints are constraints on the reader--they tell the reader what he is supposed to use as the premises of the scenario. Omega cannot cheat unless the reader interprets the description of the problem to mean that Omega is willing to cheat. And if the reader does interpret it that way, it's the reader, not Omega, who's violating the spirit of the constraints and being hyper-literal. I think that depending on the human's intentions, and assuming the human is a perfect reasoner, the conditions of the problem are contradictory. Omega can't always predict the human--it's logically impossible.
    The point here is that the question is inconsistent. It is impossible for an Omega that can predict with high accuracy to exist, as you've correctly pointed out it leads to a situation where Omega must either fail to participate, refuse to participate or cheat, which are all out of bounds of the problem.
    I don't think it's ever wise to ignore the possibility of a superintelligent AI cheating, in some manner. If we ignore that possibility, then yes, the question would be inconsistent; which implies that if the situation were to actually appear to happen, then it would be quite likely that either: * The situation has been misunderstood; or * Someone is cheating Since it is far easier for Omega, being an insane superintelligence, to cheat than it is for someone to cheat Omega, it seems likeliest that if anyone is cheating, then it is Omega. After all, Omega had and did not take the option to refuse to participate.
    I think this is the position of classical theorists on self-modifiying agents: From Rationality, Dispositions, and the Newcomb Paradox: They agree that agents who can self-modify will take one box. But they call that action "irrational". So, the debate really boils down to the definition of the term "rational" - and is not really concerned with the decision that rational agents who can self-modifiy will actually take. If my analysis here is correct, the dispute is really all about terminology.
    One factor you may not have considered: the obvious rational metastrategy is causal decision theory, and causal decision theory picks the two-box strategy.
    I don't follow. Isn't it precisely on the meta-strategy level that CDT becomes obviously irrational?
    I think what RobinZ means is that you want to choose a strategy such that having that strategy will causally yield nice things. Given that criterion, object-level CDT fails; but one uses a causal consideration to reject it.
    Key word is "obvious". If you say, "how should you solve games?", the historical answer is "using game theory", and when you say, "what does game theory imply for Newcomb's dilemma?", the historical answer is "two-box". It takes an additional insight to work out that a better metastrategy is possible, and things which take an additional insight are no longer obvious, true or no. Edit: Alternatively: When I said "metastrategy", I meant one level higher than "two-boxing" - in other words, the level of decision theory. (I'm not sure which of the two objections you were raising.)
    This is basically what I was trying to point out. :)

    Mr Eliezer, I think you've missed a few points here. However, I've probably missed more. I apologise for errors in advance.

    1. To start with, I speculate than any system of decision making consistently gives the wrong results on a specific problem. The whole point of decision theory is finding principles which usually end up with a better result. As such, you can always formulate a situation in which it gives the wrong answer: maybe one of the facts you thought you knew was incorrect, and led you astray. (At the very least, Omega may decide to reward only th
    ... (read more)
    See chapters 1-9 of this document for a more detailed treatment of the argument.
    This link is 404ing. Anyone have a copy of this?
    The current version is here. (It's Eliezer Yudkowsky (2010). Timeless Decision Theory.)

    An analogy occurs to me about "regret of rationality."

    Sometimes you hear complaints about the Geneva Convention during wartime. "We have to restrain ourselves, but our enemies fight dirty. They're at an advantage because they don't have our scruples!" Now, if you replied, "So are you advocating scrapping the Geneva Convention?" you might get the response "No way. It's a good set of rules, on balance." And I don't think this is an incoherent position: he approves of the rule, but regrets the harm it causes in thi... (read more)

    "Verbal arguments for one-boxing are easy to come by, what's hard is developing a good decision theory that one-boxes"

    First, the problem needs a couple ambiguities resolved, so we'll use three assumptions: A) You are making this decision based on a deterministic, rational philosophy (no randomization, external factors, etc. can be used to make your decision on the box) B) Omega is in fact infallible C) Getting more money is the goal (i.e. we are excluding decision-makers which would prefer to get less money, and other such absurdities)

    Changing an... (read more)

    Rather than transforming the problem in the way you did, transform it so that you move first - Omega doesn't put money in the boxes until you say which one(s) you want. As a decision problem, Newcomb's problem is rather pointless, IMHO. As a thought experiment helping us to understand the assumptions that are implicit in game theory, it could be rather useful. The thought experiment shows us that when a problem statement specifies a particular order of moves, what is really being specified is a state of knowledge at decision time. When a problem specifies that Omega moves first that is implicitly in contradiction to the claim that he knows what you will do when you move second. The implicit message is that Omega doesn't know - the explicit message is that he does. If the explicit message is to be believed, then change the move order to make the implicit message match the explicit one. However, here, many people seem to prefer to pretend that Newcomb problems constitute a decision theory problem which requires clever solution, rather than a bit of deliberate confusion constructed by violating the implicit rules of the problem genre.

    A way of thinking of this "paradox" that I've found helpful is to see the two-boxer as imagining more outcomes than there actually are. For a payoff matrix of this scenario, the two-boxer would draw four possible outcomes: $0, $1000, $1000000, and $1001000 and would try for $1000 or $1001000. But if Omega is a perfect predictor, than the two that involve it making a mistake ($0 and $1001000) are very unlikely. The one-boxer sees only the two plausible options and goes for $1000000.

    It took me a week to think about it. Then I read all the comments, and thought about it some more. And now I think I have this "problem" well in hand. I also think that, incidentally, I arrived at Eliezer's answer as well, though since he never spelled it out I can't be sure.

    To be clear - a lot of people have said that the decision depends on the problem parameters, so I'll explain just what it is I'm solving. See, Eliezer wants our decision theory to WIN. That implies that we have all the relevant information - we can think of a lot of situation... (read more)

    Let me try my own stab at a little chat with Omega. By the end of the chat I will either have 1001 K, or give up. Right now, I don’t know which. Act I Everything happens pretty much as it did in Polymeron’s dialogue, up until… Omega: Yup, that’ll work. So you’re happy with your 1000 K? Act II Whereupon I try to exploit randomness. Me: Actually, no. I’m not happy. I want the entire 1001 K. Any suggestions for outsmarting you? Omega: Nope. Me: Are you omniscient? Omega: As far as you’re concerned, yes. Your human physicists might disagree in general, but I’ve got you pretty much measured. Me: Okay, then. Wanna make a bet? I bet I can find a to get over 1000 K if I make a bet with you. You estimate your probability of being right at 100%, right? Nshepperd had a good suggestion…. Omega: I won’t play this game. Or let you play it with anyone else. I thought we’d moved past that. Me: How about I flip a fair coin to decide between B and A+B. In fact, I’ll use ’s generator using the principle to generate the outcome of a truly random coin flip. Even you can’t predict the outcome. Omega: And what do you expect to happen as a result of this (not-as-clever-as-you-think) strategy? Me: Since you can’t predict what I’ll do, hopefully you’ll fill both boxes. Then there’s a true 50% chance of me getting 1001 K. My expected payoff is 1000.5 K. Omega: That, of course, is assuming I’ll fill both boxes. Me: Oh, I’ll make you fill both boxes. I’ll bias the ’s to 50+eps% chance of one-boxing for the expected winnings of 1000.5 K – eps. Then if you want to maximize your omniscience-y-ness, you’ll have to fill both boxes. Omega: Oh, taking others’ suggestions already? Can’t think for yourself? Making edits to make it look like you’d thought of it in time? Fair enough. Attribute this one to gurgeh. As to the idea itself, I’ll disincentivize you from randomization at all. I won’t fill box B if I predict you cheating. Me: But then there’s a 50-eps% chance of proving you wron

    I wanted to consider some truly silly solution. But since taking only box A is out (and I can’t find a good reason for choosing box A, other than a vague argument based in irrationality along the lines that I’d rather not know if omniscience exists…), so I came up with this instead. I won't apologize for all the math-economics, but it might get dense.

    Omega has been correct 100 times before, right? Fully intending to take both boxes, I’ll go to each of the 100 other people. There’re 4 categories of people. Let’s assume they aren’t bound by psychology and th... (read more)

    How is there anybody in this group? Considering that all of them have $1,000,000, what convinced them to one-box in the first place such that they later changed their minds about it and regretted the decision? (Like, I guess a one-boxer could say afterwards "I bet that guy wasn't really omniscient, I should have taken the other box too, then I'd have gotten $1,001,000 instead", but why wouldn't a person who thinks that way two-box to begin with?)
    True. I only took that case into account for completeness, to cover my bases against the criticism that "not all one-boxers would be happy with their decisions." Naively, when you have a choice between 1000000.01 and 1000000.02, it's very easy to argue that the latter is the better option. To argue for the former, you would probably cite the insignificance of that cent next to the rest of 1000000.01: that eps doesn't matter, or that an extra penny in your pocket is inconvenient, or that you already have 1000000.01, so why do you need another 0.01?
    You're essentially engaging in arbitrage, taking advantage of the difference in the probabilities assigned to both boxes being full by different people. Which is one reason rational people never assign 0 probability to anything. You could just as well go to some one-boxers (who "believe P(both full) = 0") and offer them a $1 bet 10000000:1 in your favor that both boxes will be full; then offer the two-boxers whatever bet they will take "that only one box is full" that will give you more than $1 profit if you win. Thus, either way, you make a profit, and you can make however much you like just by increasing the stakes. This still doesn't actually solve newcomb's problem, though. I'd call it more of a cautionary tale against being absolutely certain. (Incidentally, since you're going into this "fully intending" to take both boxes, I'd expect both one boxers and two boxers to agree on the extremely low probability Omega is going to have filled both boxes.)
    I don't know, I feel pretty confident assigning P(A&!A)=0 :P
    "Pretty confident" is about as close to "actually 0" as the moon is (which I don't care to quantify :P).
    "Pretty confident" was also a rhetorical understatement. :P
    Do you assign 0 probability to the hypothesis that there exists something which you believe to be mathematically true which is not?
    The map is not the territory. "A&!A" would mean some fact about the world being both true and false, rather than anyone's beliefs about that fact.
    Assigning zero or nonzero probability to that assertion is having a belief about it.
    Yes, the probability is a belief, but your previous question was about something more like P(!A&P(A)=1), that is to say, an absolute belief being inconsistent with the facts. Vaniver's assertion was about the facts themselves being inconsistent with the facts, which would have a rather alarming lack of implications.
    No, P(I'm wrong about something mathematical) is 1-epsilon. P(I'm wrong about this mathematical thing) is often low- like 2%, and sometimes actually 0, like when discussing the intersection of a set and its complement. It's defined to be the empty set- there's no way that it can fail to be the empty set. I may not have complete confidence in the rest of set theory, and I may not expect that the complement of a set (or the set itself) is always well-defined, but when I limit myself to probability measures over reasonable spaces then I'm content.
    So, for some particular aspects of math, you have certainty 1-epsilon, where epsilon is exactly zero? What you are really doing is making the claim "Given that what I know about mathematics is correct, then the intersection of a set and its complement is the empty set."
    I was interpreting "something" as "at least one thing." Almost surely my understanding of mathematics as a whole is incorrect somewhere, but there are a handful of mathematical statements that I believe with complete metaphysical certitude. "Correct" is an unclear word, here. Suppose I start off with a handful of axioms. What is the probability that one of the axioms is true / correct? In the context of that system, 1, since it's the starting point. Now, the axioms might not be useful or relevant to reality, and the axioms may conflict and thus the system isn't internally consistent (i.e. statements having probability 0 and 1 simultaneously). And so the geometer who is only 1-epsilon sure that Euclid's axioms describe the real world will be able to update gracefully when presented with evidence that real space is curved, even though they retain the same confidence in their Euclidean proofs (as they apply to abstract concepts). Basically, I only agree with this post when it comes to statements about which uncertainty is reasonable. If you require 1-epsilon certainty for anything, even P(A|A), then you break the math of probability.
    Yes, nshepperd, my assumption is that P << 0.5, something in the 0.0001 to 0.01 range. Besides, arbitrage would still be possible if some people estimated P=0.01 and others P=0.0001, only the solution would be messier than what I'd ever want to do casually. Besides, if I were unconstrained by the bets I could make (I'd tried to work with a cap before), that would make making profits even easier. I wasn't exactly trying to solve the problem, only to find a "naively rational" workaround (using the same naive rationality that leads prisoners to rat each other out in PD). When you're saying that this doesn't solve Newcomb's problem, what do you expect the solution to actually entail?
    Yes, arbitrage is possible pretty much whenever people's probabilities disagree to any significant degree. Setting P = 0 just lets you take it to absurd levels (eg. put up no stake at all, and it's still a "fair bet"). Maximizing the money found upon opening the box(es) you have selected. If you like, replace the money with cures for cancer with differing probabilities of working, or machines with differing probabilities of being a halting oracle, or something else you can't get by exploiting other humans.

    1) I would one-box. Here's where I think the standard two-boxer argument breaks down. It's the idea of making a decision. The two-boxer idea is that once the boxes have been fixed the course of action that makes the most money is taking both boxes. Unless there is reverse causality going on here, I don't think that anyone disputes this. If at that moment you could make a choice totally independently of everything leading up to that point you would two-box. Unfortunately, the very existence of Omega implies that such a feat is impossible.

    2) A mildly s... (read more)

    dankane, Eliezer answered your question in this comment, and maybe somewhere else, too, that I don't yet know of.
    If he wasn't really talking about infinities, how would you parse this comment (the living forever part): "There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever." At very least this should imply that for every N there is an f(N) so that he would rather have a 50% chance of living f(N) years and a 50% chance of dying instantly than having a 100% chance of living for N years. We could then consider the game where if he is going to live for N years he is repeatedly offered the chance to instead live f(N) years with 50% probability and 0 years with 50% probability. Taking the bet n+1 times clearly does better than taking it n times, but the strategy "take the bet until you lose" guarantees him a very short life expectancy. If your utility function is unbounded you can run into paradoxes like this.

    Actually I take it back. I think that what I would do depends on what I know of how Omega functions (exactly what evidence lead me to believe that he was good at predicting this).

    Omega #1: (and I think this one is the most plausible) You are given a multiple choice personality test (not knowing what's about to happen). You are then told that you are in a Newcomb situation and that Omega's prediction is based on your test answers (maybe they'll even show you Omega's code after the test is over). Here I'll two-box. If I am punished I am not being punish... (read more)

    The first case directly contradicts the specifications of the problem, since the idea then becomes to imagine you were the sort of person who would one-box and answer like that, then two box. This might not work for everyone, but a sufficiently clever agent should manage it. If you are imagining a personality test undertaken in secret, or before you knew you were facing Newcomb's problem, and stating you would two-box, then it seems like you one-box when it is absolutely certain that omega is right, but two-box if you can think of some way (however unlikely) that he might be wrong. If you don't see the problem with this then I suggest you read some of the sequence posts about absolute certainty.
    In the first case, I image the test undertaken in secret. Or more realistically Omega measures these personality traits from listening to my conversations, or reading things I post online. I don't decide based on whether there is a possibility that Omega is wrong. #2 can certainly be wrong (for example if I decide based on coin flip) and even #3 can probably mess up. My point is that in case #1 the argument from the post no longer works. If I two-boxed and didn't get $1M, I might envy another person for their personality traits (which correlate with one-boxing), but not their decision to one-box. I think what I am trying to do is split Omega's decision procedure into cases where either: * His prediction is clearly caused by my decision (so I should one-box) * His prediction is not caused by my decision (and so I can two-box without regretting my choice) (#2 is a special case where I try to be clever.)
    Okay, I misunderstood you. Even now, I think I would still one-box in case#1. For one thing, it is clearly in my interests, thinking about the problem in advance, to resolve to do so, since the personality test will reveal this fact and I will get the million. Would you agree with me that far? If so, how do you handle the problem that you seem to be making different decisions at different times, without receiving any new information in between.
    Do you really think that merely deciding to one-box in such a situation would change your personality in a way that gets picked up by the test? If it does, do you want to modify your personality in a measurable way just so that you can win if you happen to run into a Newcomb problem? Suppose for example it had been determined empirically that whether or not one was religious correlated well with the number of boxes you took. This could then be one of the things that the personality test measures. Are you saying that a precommitment would change your religious beliefs, or that you would change them in addition to deciding to one-box (in which case, why are you changing the latter at all)? The point in case 1 is that they are not making a direct measurement of your decision. They are merely measuring external factors so that for 99% of people these factors agree with their decision (I think that this is implausible, but not significantly more implausible than the existence of Omega in the first place). It seems to me very unlikely that just changing your mind on whether you should one-box would also automatically change these other factors. And if it does, do you necessarily want to be messing around with your personality just to win this game that will almost certainly never come up?
    If merely deciding to one-box is not picked up by the test, and does not offer even a slight increase in the probability that the money is there (even 51% as opposed to 50% would be enough) then the test is not very good, in which case I would two-box. However, this seems to contradict the stated fact the Omega is in fact a very good predictor of decisions. As a general principle, I am most definitely interested in modifying my personality to increase the number of situations in which I win. If I wasn't, I probably wouldn't be on LW. The religion example is a strawman, as it seems clear that applying the modification "believe in God" will cause me to do worse in many other much more common situations, whereas "one-box in Newcomb-type dilemma's" doesn't seem likely to have many side effects. If Omega really is just measuring external factor's, then how do you know he won't pick up on my decision to always one-box. The decision was not made in a vacuum, it was caused by my personality, my style of thinking and my level of intelligence, all of which are things hat any reasonably competent predictor should pick up on. As long as the test is reasonably good, I will still my million with a higher probability, and that's all that really matters to me.
    I don't think that you change of just that decision would be picked up on a personality test. Your changing that decision is unlikely to change how you answer questions not directly relating to Newcomb's problem. The test would pick up your style of thinking that lead you to this decision, but making the decision differently would not change your style of thinking. Perhaps an example that illustrates my point even better: Omega #1.1: Bases his prediction on a genetic test. Now I agree that it is unlikely that this will get 99% accuracy, but I think it could plausibly obtain, say, 60% accuracy, which shouldn't really change the issue at hand. Remember that Omega does not need to measure things that cause you to decide one way or another, he just needs to measure things that have a positive correlation with it. As for modifying your personality... Should I really believe that you believe that arguments that you are making here, or are you just worried that you are going to be in this situation and that Omega will base his prediction on your posts?
    Good point with the genetic test argument, in that situation I probably would two-box. The same might apply to any sufficiently poor personality test, or to a version of Omega that bases his decision of the posts I make on Less Wrong (although I think if my sole reason for being here was signalling my willingness to make certain choices in certain dilemma's I could probably find better ways to do it). I usually imagine Omega does better than that, and that his methods are at least as sophisticated as figuring out how I make decisions, then applying that algorithm to the problem at hand (the source of this assumption is that the first time I saw the problem Omega was a supercomputer that scanned people's brains). As for the personality modification thing, I really don't see what you find so implausible about the idea that I'm not attached to my flaws, and would eliminate them if I had the chance.
    I agree that the standard interpretation of Omega generally involves brain scans. But there is still a difference between running a simulation (Omega #2), or checking for relevant correlating personality traits. The later I would claim is at least somewhat analogous to genetic testing, though admittedly the case is somewhat murkier. I guess perhaps the Omega that is most in the spirit of the question is where he does a brain scan and searches for your cached answer of "this is what I do in Newcomb problems". As for personality modification, I don't see why changing my stored values for how to behave in Newcomb situations would change how I behave in non-Newcomb situations. I also don't see why these changes would necessarily be an improvement.
    "I don't see why changing my stored values for how to behave in Newcomb situations would change how I behave in non-Newcomb situations." It wouldn't, that's the point. But it would improve your performance in Newcomb situations, so there's no downside (for an example of a newcomb type paradox which could happen in the real world, see Parfit's hitch-hiker, given that I am not a perfect liar I would not consider it too unlikely that I will face a situation of that general type (if not that exact situation) at some point in my life).
    My point was that if it didn't change your behavior in non-Newcomb situations, no reasonable version of Omega #1 (or really any Omega that does not use either brain scans or lie detection could tell the difference). As for changing my actions in the case of Parfit's hitch-hiker, say that the chances of actually running into this situation (with someone who can actually lie detect and in a situation with no third alternatives, and where my internal sense of fairness wouldn't just cause me to give him the $100 anyway) is say 10^-9. This means that changing my behavior would save me an expected say 3 seconds of life. So if you have a way that I can actually precommit myself that takes less than 3 seconds to do, I'm all ears.
    It wouldn't have to be that exact situation. In fact, it is applicable in any situation where you need to make a promise to someone who has a reasonable chance of spotting if you lie (I don't know about you but I often get caught out when I lie), and while you prefer following through on the promise to not making it, you also prefer going back on the promise to following through on it, (technically they need to have a good enough chance of spotting you, with "good enough" determined by your relative preferences). That's quite a generic situation, and I would estimate at least 10% probability that you encounter it at some point, although the stakes will hopefully be lower than your life.
    Perhaps. Though I believe that in the vast majority of these cases my internal (and perhaps irrational) sense of fairness would cause me to keep my word anyway.
    How about this version of Omega (and this is one that I think could actually be implemented to be 90% accurate). First off, box A is painted with pictures of snakes and box B with pictures of bananas. Omega's prediction procedure is (and you are told this by the people running the experiment) that if you are a human he predicts that you two-box and if you are a chimpanzee, he predicts that you one-box. I don't think that 10% of people would give up $1000 to prove Omega wrong, and if you think so, why not make it $10^6 and $10^9 instead of $10^3 and $10^6. I feel like this version satisfies the assumptions of the problem and makes it clear that you should two-box in this situation. Therefore any claims that one-boxing is the correct solution need to at least be qualified by extra assumptions about how Omega operates.
    In this version Omega may be predicting decision's in general with some accuracy, but it does not seem like he is predicting mine. So it appears there are cases where I two-box. I think in general my specification of a Newcomb-type problem, has two requirements: An outside observer who observed me to two-box would predict with high-probability that the money is not there. An outside observer who observed me to one-box would predict with high-probability that the money is there. The above version of the problem clearly does not meet the second requirement. If this is what you meant by your statement that the problem is ambiguous, then I agree. This is one of the reasons I favour a formulation involving a brain-scanner rather than a nebulous godlike entity, since it seems more useful to focus on the particularly paradoxical cases rather than the easy ones.

    "the dominant consensus in modern decision theory is that one should two-box...there's a common attitude that verbal arguments for one-boxing are easy to come by, what's hard is developing a good decision theory that one-boxes"

    This may be more a statement about the relevance and utility of decision theory itself as a field (or lack thereof) than the difficulty of the problem, but it is at least philosophically intriguing.

    From a physical and computational perspective, there is no paradox, and one need not invoke backwards causality, 'pre-commitmen... (read more)

    Upon reading this, I immediately went,

    "Well, General Relativity includes solutions that have closed timelike curves, and I certainly am not in any position to rule out the possibility of communication by such. So I have no actual reason to rule out the possibility that which strategy I choose will, after I make my decision, be communicated to Omega in my past and then the boxes filled accordingly. So I better one-box in order to choose the closed timelike loop where Omega fills the box."

    I understand, looking at Wikipedia, that in Nozick's formu... (read more)

    The "no backwards causality" argument seems like a case of conflation of correlation and causation. Your decision doesn't retroactively cause Omega to fill the boxes in a certain way; some prior state of the world causes your thought processes and Omega's prediction, and the correlation is exactly or almost exactly 1.

    EDIT: Correlation coefficients don't work like that, but whatever. You get what I mean.


    The "no backwards causality" argument seems like a case of conflation of correlation and causation. Your decision doesn't retroactively cause Omega to fill the boxes in a certain way; some prior state of the world causes your thought processes and Omega's prediction, and the correlation is exactly or almost exactly 1.

    The original description of the problem doesn't mention if you know of Omega's strategy for deciding what to place in box B, or their success history in predicting this outcome - which is obviously a very important factor.

    If you know these things, then the only rational choice, obviously and by a huge margin, is to pick only box B.

    If you don't know anything other than box B may or may not contain a million dollars, and you have no reasons to believe that it's unlikely, like in the lottery, then the only rational decision is to take both. This also seems to... (read more)

    It looks like you just restated the "paradox" - using one argument, it is "obvious" to pick B and using another argument, it is "obvious" to pick both. Also, in general, do try to avoid saying something is "obvious". It usually throws a lot of complexity and potential faults into a black box and worsens your chances of uncovering those faults by intimidating people.

    You are betting a positive extra payout of $1,000 against a net loss of -$999,000 that there are no Black Swans[1] at all in this situation.

    Given that you already have 100 points of evidence that taking Box A makes Box B empty (added to the evidence that Omega is more intelligent than you). I'd say that's a Bad Bet to make.

    Given the amount of uncertainty in the world, choosing Box B instead of trying to "beat the system" seems like the rational step to me.

    Edit I've given the Math in a comment below to show how to calculate when to make either dec... (read more)

    Edit: Never mind, my comment resulted from a confusion.
    I'm not sure what you are implying with this link - can you please expand? Are you saying that I'm choosing a least convenient possible world (and if so, how and what) or that 2-boxers are doing so?
    You are finding technical flaws that are not essential to the intended sense of the thought experiment. Instead of making it uninteresting because of the potential flaws, make the thought experiment stronger by considering the case where these flaws are fixed.
    Sorry, your comment was confusing and I didn't properly concentrate on what you meant, so giving the LCPW link was a mistake, it doesn't seem to apply.
    No problem. I've expanded with the math explaining what I mean, hopefully that makes it less confusing what I was aiming at.
    So working the math on that Let P(BS) = probability of a Black Swan being involved This makes the average payout work out to: 1-Box = $1,000,000 2-Box = $1,001,000 (1 - P(BS)) + $1,000 P(BS) Now it seems to be that the average 2-boxer is assuming that P(BS) = 0, which would make the 2-Box solution always == $1,001,000 which would, of course, always beat the 1-box solution. and maybe in this toy-problem, they're right to assume P(BS) = 0 But IRL that's almost never the case - after all, 0 is not a probability yes? So assume that P(BS) is non-zero. t what point would it be worth it to choose the 1-Box solution and what point the 2-Box solution? Lets run the math: 1,000,000 = 1,001,000(1-x) + 1000x = 1001000 - 1001000x + 1000x = 1001000 - (1002000x) => 1000000 - 1001000 = -1002000x => x = -1000/-100200 => x = 0.000998004 So, the estimated probability of Black Swan existing only has to be greater than 0.0998% for the 1-Box solution to have a greater expected payout and therefore the 1-Box option is the more rational::Bayesian choice OTOH, if you can guarantee that P(BS) is less than 0.0998%, then the rational choice is to 2-Box.

    How would Newcomb's problem look like in the physical world, taking quantum physics into account? Specifically, would Omega need to know quantum physics in order to predict my decision on "to one box or not to one box"?

    To simplify the picture, imagine that Omega has a variable with it that can be either in the state A+B or B and which is expected to correlate with my decision and therefore serves to "predict" me. Omega runs some physical process to arrive at the contents of this variable. I'm assuming that "to predict" means &... (read more)

    ...if you build an AI that two-boxes on Newcomb's Problem, it will self-modify to one-box on Newcomb's Problem, if the AI considers in advance that it might face such a situation. Agents with free access to their own source code have access to a cheap method of precommitment.


    But what does an agent with a disposition generally-well-suited to Newcomblike problems look like? Can this be formally specified?


    Rational agents should WIN.

    It seems to me that if all that is true, and you want to build a Friendly AI, then the rational thing to do he... (read more)

    Eliezer says elsewhere that current decision theory doesn't let us prove a self-modifying AI would choose to keep the goals we program into it. He wants to develop a proof before even starting work on the AI.
    It's easy to contrive situations where a self-modifying AI would choose not to keep the goals programmed into it, even without precommitment issues. Just contrive the circumstances so it gets paid to change. Unless there's something wrong with the argument there, TDT etc. won't be enough to ensure that the goals are kept.
    4Eliezer Yudkowsky12y
    Causal decision theorists don't self-modify to timeless decision theorists. If you get the decision theory wrong, you can't rely on it repairing itself.

    You said:

    Causal decision theorists don't self-modify to timeless decision theorists. If you get the decision theory wrong, you can't rely on it repairing itself.

    but you also said:

    ...if you build an AI that two-boxes on Newcomb's Problem, it will self-modify to one-box on Newcomb's Problem, if the AI considers in advance that it might face such a situation.

    I can envision several possibilities:

    • Perhaps you changed your mind and presently disagree with one of the above two statements.
    • Perhaps you didn't mean a causal AI in the second quote. In that case I have no idea what you meant.
    • Perhaps Newcomb's problem is the wrong example, and there's some other example motivating TDT that a self-modifying causal agent would deal with incorrectly.
    • Perhaps you have a model of causal decision theory that makes self-modification impossible in principle. That would make your first statement above true, in a useless sort of way, so I hope you didn't mean that.

    Would you like to clarify?

    Causal decision theorists self-modify to one-box on Newcomb's Problem with Omegas that looked at their source code after the self-modification took place; i.e., if the causal decision theorist self-modifies at 7am, it will self-modify to one-box with Omegas that looked at the code after 7am and two-box otherwise. This is not only ugly but also has worse implications for e.g. meeting an alien AI who wants to cooperate with you, or worse, an alien AI that is trying to blackmail you.

    Bad decision theories don't necessarily self-repair correctly.

    And in general, every time you throw up your hands in the air and say, "I don't know how to solve this problem, nor do I understand the exact structure of the calculation my computer program will perform in the course of solving this problem, nor can I state a mathematically precise meta-question, but I'm going to rely on the AI solving it for me 'cause it's supposed to be super-smart," you may very possibly be about to screw up really damned hard. I mean, that's what Eliezer-1999 thought you could say about "morality".

    Okay, thanks for confirming that Newcomb's problem is a relevant motivating example here. I'm not saying that. I'm saying that self-modification solves the problem, assuming the CDT agent moves first, and that it seems simple enough that we can check that a not-very-smart AI solves it correctly on toy examples. If I get around to attempting that, I'll post to LessWrong. Assuming the CDT agent moves first seems reasonable. I have no clue whether or when Omega is going to show up, so I feel no need to second-guess the AI about that schedule. (Quoting out of order) As you know, we can define a causal decision theory agent in one line of math. I don't know a way to do that for TDT. Do you? If TDT could be concisely described, I'd agree that it's the less ugly alternative. I'm failing to suspend disbelief here. Do you have motivating examples for TDT that seem likely to happen before Kurzweil's schedule for the Singularity causes us to either win or lose the game?
    2Wei Dai12y
    If you appreciate simplicity/elegance, I suggest looking into UDT. UDT says that when you're making a choice, you're deciding the output of a particular computation, and the consequences of any given choice are just the logical consequences of that computation having that output. CDT in contrast doesn't answer the question "what am I actually deciding when I make a decision?" nor does it answer "what are the consequences of any particular choice?" even in principle. CDT can only be described in one line of math because the answer to the latter question has to be provided to it via an external parameter.
    Thanks, I'll have a look at UDT. I certainly agree there.
    I'm reasonably sure Eliezer meant implications for the would-be friendly AI meeting alien AIs. That could happen at any time in the remaining life span of the universe.
    Maybe this one: "Argmax[A in Actions] in SumO in Outcomes*P(this computation yields A []-> O|rest of universe)" From this post.
    Why not? A causal decision theorist can have an accurate abstract understanding of both TDT and CDT and can calculate the expected utility of applying either. If TDT produces a better expected outcome in general then it seems like self modifying to become a TDT agent is the correct decision to make. Is there some restriction or injunction assumed to be in place with respect to decision algorithm implementation? Thinking about it for a a few minutes: It would seem that the CDT agent will reliably update away from CDT but that the new algorithm will be neither CDT or TDT (and not UDT either). It will be able to cooperate with agents when there has been some sort causal entanglement between the modified source code and the other agent but not able to cooperate with complete strangers. The resultant decision algorithm is enough of an attractor that it deserves a name of its own. Does it have one?
    2Paul Crowley12y
    Surely the important thing is that it will self-modify to whatever decision theory has the best consequences? The new algorithm will not exactly be TDT, because it won't try to change decisions that have already been made the way TDT does. In particular this means that there's no risk from Roko's basilisk. Disclaimer: I'm not very confident of anything I say about decision theory.
    7Eliezer Yudkowsky12y
    Doesn't have a name as far as I know. But I'm not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch? Maybe, now that I think about it, but even so, does it matter? But it will calculate that expected value using CDT!expectation, meaning that it won't see how self-modifying to be a timeless decision theorist could possibly affect what's already in the box, etcetera.
    1Paul Crowley12y
    Is that really so bad, if it takes the state of the world at the point before it self-modifies as an unchangeable given, and self-modifies to a decision theory that only considers states from that point on as changeable by its decision theory? For one thing, doesn't that avoid Roko's basilisk?
    3Wei Dai12y
    If you do that, you'd be vulnerable to extortion from any other AIs that happen to be created earlier in time and can prove their source code.
    1Paul Crowley12y
    I'm inclined to think that in most scenarios the first AGI wins anyway. And leaving solving decision theory to the AGI could mean you get to build it earlier.
    3Wei Dai12y
    I was thinking of meeting alien AIs, post-Singularity. Huh? I thought we were supposed to be the good guys here? ;-) But seriously, "sacrifice safety for speed" is the "defect" option in the game of "let's build AGI". I'm not sure how to get the C/C outcome (or rather C/C/C/...), but it seems too early to start talking about defecting already. Besides, CDT is not well defined enough that you can implement it even if you wanted to. I think if you were forced to implement a "good enough" decision theory and hope for the best, you'd pick UDT at this point. (UDT is also missing a big chunk from its specifications, namely the "math intuition module" but I think that problem has to be solved anyway. It's hard to see how an AGI can get very far without being able to deal with logical/mathematical uncertainty.)
    Really? That's surprising. My assumption had been that CDT would be much simpler to implement - but just give undesirable outcomes in whole classes of circumstance.
    5Wei Dai12y
    CDT uses a "causal probability function" to evaluate the expected utilities of various choices, where this causal probability function is different from the epistemic probability function you use to update beliefs. (In EDT they are one and the same.) There is no agreement amongst CDT theorists how to formulate this function, and I'm not aware of any specific proposal that can be straightforwardly implemented. For more details see James Joyce's The foundations of causal decision theory.
    I understand AIXI reasonably well and had assumed it was a specific implementation of CDT, perhaps with some tweaks so the reward values are generated internally instead of being observed in the environment. Perhaps AIXI isn't close to an implementation of CDT, perhaps it's perceived as not specific or straightforward enough, or perhaps it's not counted as an implementation. Why isn't AIXI a counterexample?
    2Wei Dai12y
    You may be right that AIXI can be thought of as an instance of CDT. Hutter himself cites "sequential decision theory" from a 1957 paper which certainly predates CDT, but CDT is general enough that SDT could probably fit into its formalism. (Like EDT can be considered an instance of CDT with the causal probability function set to be the same as the epistemic probability function.) I guess I hadn't considered AIXI as a serious candidate due to its other major problems.
    Four problems are listed there. The first one is the claim that AIXI wouldn't have a proper understanding of its body because its thoughts are defined mathematically. This is just wrong, IMO; my refutation, for a machine that's similar enough to AIXI for this issue to work the same, is here. Nobody has engaged me in serious conversation about that, so I don't know how well it will stand up. (If I'm right on this, then I've seen Eliezer, Tim Tyler, and you make the same error. What other false consensuses do we have?) The second one is fixed if we do the tweak I mentioned in the grandparent of this comment. If you take the fix described above for the second one, what's left of the third one is the claim that instantaneous human (or AI) experience is too nuanced to fit in a single cell of a Turing machine. According to the original paper, page 8, the symbols on the reward tape are drawn from an alphabet R of arbitrary but fixed size. All you need is a very large alphabet and this one goes away. I agree with the facts asserted in Tyler's fourth problem, but I do not agree that it is a problem. He's saying that Kolmogorov complexity is ill-defined because the programming language used is undefined. I agree that rational agents might disagree on priors because they're using different programming languages to represent their explanations. In general, a problem may have multiple solutions. Practical solutions to the problems we're faced with will require making indefensible arbitrary choices of one potential solution over another. Picking the programming language for priors is going to be one of those choices.
    FYI, generating reward values internally - instead of them being observed in the environment - makes no difference whatsoever to the wirehead problem. AIXI digging into its brains with its own mining claws is quite plausible. It won't reason as you suggest - since it has no idea that it is instantiated in the real world. So, its exploratory mining claws may plunge in. Hopefully it will get suitably negatively reinforced for that - though much will depend on which part of its brain it causes damage too. It could find that ripping out its own inhibition circuits is very rewarding. A larger set of symbols for rewards makes no difference - since the reward signal is a scalar. If you compare with an animal, that has millions of pain sensors that operate in parallel. The animal is onto something there - something to do with a-priori knowledge about the common causes of pain. Having lots of pain sensors has positive aspects - e.g. it saves you experimenting to figure out what hurts. As for the reference machine issue, I do say: "This problem is also not very serious." Not very serious unless you are making claims about your agent being "the most intelligent unbiased agent possible". Then this kind of thing starts to make a difference...
    You can encode 16 64 bit integers in a 1024 bit integer. The scalar/parallel distinction is bogus. (Edit: I original wrote "5 32 bit integers" when I meant "2**5 32 bit integers". Changed to "16 64 bit integers" because "32 32 bit integers" looked too much like a typo.) Strawman argument. The only claim made is that it's the most intelligent up to a constant factor, and a bunch of other conditions are thrown in. When Hutter's involved, you can bet that some of the constant factors are large compared to the size of the universe.
    Er, not if you are adding the rewards together and maximising the results, you can't! That is exactly what happens to the rewards used by AIXI. Actually Hutter says this sort of thing all over the place (I was quoting him above) - and it seems pretty irritating and misleading to me. I'm not saying the claims he makes in the fine print are wrong, but rather that the marketing headlines are misleading.
    You're right there, I'm confusing AIXI with another design I've been working with in a similar idiom. For AIXI to work, you have to combine together all the environmental stuff and compute a utility, make the code for doing the combining part of the environment (not the AI), and then use that resulting utility as the input to AIXI.
    3Wei Dai12y
    I don't see how your refutation applies to AIXI. Let me just try to explain in detail why I think AIXI will not properly protect its body. Consider an AIXI that arises in a simple universe, i.e., one computed by a short program P. AIXI has a probability distribution not over universes, but instead over environments where an environment is a TM whose output tape is AIXI's input tape and whose input tape is AIXI's output tape. What's the simplest environment that fits AIXI's past inputs/outputs? Presumably it's E = P plus some additional pieces of code that injects E's inputs into where AIXI's physical output ports are located in the universe (that is, overrides the universe's natural evolution using E's inputs), and extracts E's outputs from where AIXI's physical input ports are located. What happens when AIXI considers an action that destroys its physical body in the universe computed by P? As long as the input/output ports are not also destroyed, AIXI would expect that the environment E (with its "supernatural" injection/extraction code) will continue to receive its outputs and provide it with inputs. Does that make sense?
    (Responding out of order) Yes, but it makes some unreasonable assumptions. An implementation of AIXI would be fairly complex. If P is too simple, then AIXI could not really have a body in the universe, so it would be correct in guessing that some irregularity in the laws of physics was causing its behaviors to be spliced into the behavior of the world. However, if AIXI has observed enough of the inner workings of other similar machines, or enough of the laws of physics in general, or enough of its own inner workings, the simplest model will be that AIXI's outputs really do emerge from the laws of physics in the real universe, since we are assuming that that is indeed the case and that Kolmogorov induction eventually works. At that point, imagining that AIXI's behaviors are a consequence of a bunch of exceptions to the laws of physics is just extra complexity and won't be part of the simplest hypothesis. It will be part of some less likely hypotheses, and the AI would have to take that risk into account when deciding whether to self-improve.
    0Wei Dai12y
    Tim, I think you're probably not getting my point about the distinction between our concept of a computable universe, and AIXI's formal concept of a computable environment. AIXI requires that the environment be a TM whose inputs match AIXI's past outputs and whose outputs match AIXI's past inputs. A candidate environment must have the additional code to inject/extract those inputs/outputs and place them on the input/output tapes, or AIXI will exclude it from its expected utility calculations.
    I agree that the candidate environment will need to have code to handle the inputs. However, if the candidate environment can compute the outputs on its own, without needing to be given the AI's outputs, the candidate environment does not need code to inject the AI's outputs into it. Even if the AI can only partially predict its own behavior based on the behavior of the hardware it observes in the world, it can use that information to more efficiently encode its outputs in the candidate environment, so it can have some understanding of its position in the world even without being able to perfectly predict its own behavior from first principles. If the AI manages to destroy itself, it will expect its outputs to be disconnected from the world and have no consequences, since anything else would violate its expectations about the laws of physics. This back-and-forth appears to be useless. I should probably do some Python experiments and we then can change this from a debate to a programming problem, which would be much more pleasant.
    2Wei Dai12y
    If a candidate environment has no special code to inject AIXI's outputs, then when AIXI computes expected utilities, it will find that all actions have equal utility in that environment, so that environment will play no role in its decisions. Ok, but try not to destroy the world while you're at it. :) Also, please take a closer look at UDT first. Again, I think there's a strong possibility that you'll end up thinking "why did I waste my time defending CDT/AIXI?"
    Thankyou for the reference, and the explanation. I am prompted to ask myself a question analogous to the one Eliezer recently asked: Is it worth my while exploring the details of CDT formalization beyond just the page you linked to? There seems to be some advantage to understanding the details and conventions of how such concepts are described. At the same time revising CDT thinking in too much detail may eliminate some entirely justifiable confusion as to why anyone would think it is a good idea! "Causal Expected Utiluty"? "Causal Tendencies"? What the? I only care about what will get me the best outcome!
    1Wei Dai12y
    Probably not. I only learned it by accident myself. I had come up with a proto-UDT that was motivated purely by anthropic reasoning paradoxes (as opposed to Newcomb-type problems like CDT and TDT), and wanted to learn how existing decision theories were formalized so I could do something similar. James Joyce's book was the most prominent such book available at the time. ETA: Sorry, I think the above is probably not entirely clear or helpful. It's a bit hard for me to put myself in your position and try to figure out what may or may not be worthwhile for you. The fact is that Joyce's book is the decision theory book I read, and quite possibly it influenced me more than I realize, or is more useful for understanding the motivation for or the formulation of UDT than I think. It couldn't hurt to grab a copy of it and read a few chapters to see how useful it is to you.
    Thanks for the edit/update. For reference it may be worthwhile to make such additions as a new comment, either as a reply to yourself or the parent. It was only by chance that I spotted the new part!
    What pre-singularity actions are you worried about them taking? What I was thinking was that a CDT-seeded AI might actually be safer precisely because it won't try to change pre-Singularity events, and if it's first the new decision theory will be in place in time for any post-Singularity events. That's surprising to me - what should I read in order to understand this point better?
    1Paul Crowley12y
    What pre-singularity actions are you worried about them taking? What I was thinking was that a CDT-seeded AI might actually be safer precisely because it won't try to change pre-Singularity events, and if it's first the new decision theory will be in place in time for any post-Singularity events. That's surprising to me - what should I read in order to understand this point better? EDIT: strike that, you answer that above.
    2Wei Dai12y
    They could modify themselves so that if they ever encounter a CDT-descended AI they'll start a war (even if it means mutual destruction) unless the CDT-descended AI gives them 99% of its resources.
    They could also modify themselves to make the analogous threat if they encounter a UDT-descended AI, or a descendant of an AI designed by TIm Freeman, or a descendant of an AI designed by Wei Dai, or a descendant of an AI designed using ideas mentioned on LessWrong. I would hope that any of those AI's would hand over 99% of their resources if the extortionist could prove its source code and prove that war would be worse. I assume you're saying that CDT is special in this regard. How is it special? (Thanks for the pointer to the James Joyce book, I'll have a look at it.)
    9Wei Dai12y
    If the alien AI computes the expected utility of "provably modify myself to start a war against CDT-AI unless it gives me 99% of its resources", it's certain to get a high value, whereas if it computes the expected utility of "provably modify myself to start a war against UDT-AI unless it gives me 99% of its resources" it might possibly get a low value (not sure because UDT isn't fully specified), because the UDT-AI, when choosing what to do when faced with this kind of threat, would take into account the logical correlation between its decision and the alien AI's prediction of its decision.
    Well, that's plausible. I'll have to work through some UDT examples to understand fully. What model do you have of how entity X can prove to entity Y that X is running specific source code? The proof that I can imagine is entity Y gives some secure hardware Z to X, and then X allows Z to observe the process of X self-modifying to run the specified source code, and then X gives the secure hardware back to Y. Both X and Y can observe the creation of Z, so Y can know that it's secure and X can know that it's a passive observer rather than a bomb or something. This model breaks the scenario, since a CDT playing the role of Y could self-modify any time before it hands over Z and play the game competently. Now, if there's some way for X to create proofs of X's source code that will be convincing to Y without giving advance notice to Y, I can imagine a problem for Y here. Does anyone know how to do that? (I acknowledge that if nobody knows how to do that, that means we don't know how to do that, not that it can't be done.) Hmm, this explains my aversion to knowing the details of what other people are thinking. It can put me at a disadvantage in negotiations unless I am able to lie convincingly and say I do not know.
    3Wei Dai12y
    I think I''ll stop here for now, because you already seem intrigued enough to want to learn about UDT in detail. I'm guessing that once you do, you won't be so motivated to think up reasons why CDT isn't really so bad. :) Let me know if that turns out not to be the case though.
    1Wei Dai12y
    On second thought, I should answer this question because it's of independent interest. If Y is sufficiently powerful, it may be able to deduce the laws of physics and the initial conditions of the universe, and then obtain X's source code by simulating the universe up to when X is created. Note that Y may do this not because it wants to know X's source code in some anthropomorphic sense, but simply due to how its decision-making algorithm works.
    Unless there have been some specific assumptions made about the universe that will not work. Simulating the entire universe does not tell Y which part of the universe it inhabits. It will give Y a set of possible parts of the universe which match Y's observations. While the simulation strategy will allow the best possible prediction about what X's source code is given what Y already knows it does not give evidence to Y that it didn't already have.
    0Wei Dai12y
    You're right, the model assumes that we live in a universe such that superintelligent AIs would "naturally" have enough evidence to infer the source code of other AIs. (That seems quite plausible, although by no means certain, to me.) Also, since this is a thread about the relative merits of CDT, I should point out that there are some games in which CDT seems to win relative to TDT or UDT, which is a puzzle that is still open.
    It's an interesting problem, but my impression when reading was somewhat similar to that of Eliezer in the replies. At the core it is the question of "How do you deal with constructs made by other agents?" I don't think TDT has any particular weakness there.
    Quantum mechanics seems to be pretty clear that true random number generators are available, and probably happen naturally. I don't understand why you consider that scenario probable enough to be worth talking about.
    Do you have an intuition as to how it would do this without contradicting itself? I tried to ask a similar question but got it wrong in the first draft and afaict did not receive an answer to the relevant part. I just want to know if my own intuition fails in the obvious way.
    It is useful to separate in one's mind the difference between on one hand being able to One Box and cooperate in PD with agents that you know well (shared source code) and on the other hand not firing on Baby Eaters after they have already chosen not to fire on you. This is especially the case when first grappling the subject. (Could you confirm, by the way, that Akon's decision in that particular paragraph or two is approximately what TDT would suggest?) The above is particularly relevant because the "have access to each other's source code" is such a useful intuition pump when grappling with or explaining the solutions to many of the relevant decision problems. It is useful to be able to draw a line on just how far the source code metaphor can take you. There is also something distasteful about making comparisons to a decision theory that isn't even implicitly stable under self modification. A CDT agent will change to CDT++ unless there is an additional flaw in the agent beyond the poor decision making strategy. If I create a CDT agent, give it time to think and then give it Newcomb's problem it will One Box (and also no longer be a CDT agent). It is the errors in the agent that still remain after that time that need TDT or UDT to fix. *nod* This is just the 'new rules starting now' option. What the CDT agent does when it wakes up in an empty, boring room and does some introspection.
    The CDT is making a decision about whether to self-modify even before it meets the alien, based on its expectation of meeting the alien. How does CDT!expectation differ from Eliezer!expectation before we meet the alien?
    Yes, because there are lemmas you can prove about (some) decision theory problems which imply that CDT and UDT give the same output. For example, CDT works if there is exists a total ordering over inputs given to the strategy, common to all execution histories, such that the world program invokes the strategy only with increasing, non-repeating inputs on that ordering. There are (relatively) easy algorithms for these cases. CDT in general is then a matter of applying a theorem when one of its preconditions doesn't hold, which is one of the most common math mistakes ever.
    Yes, for reasons of game theory and of practical singularity strategy. Game theory, because things in Everett branches that are 'closest' to us might be the ones it's most important to be able to interact with, since they're easier to simulate and their preferences are more likely to have interesting overlap with ours. Knowing very roughly what to expect from our neighbors is useful. And singularity strategy, because if you can show that architectures like AIXI-tl have some non-negligible chance of converging to whatever an FAI would have converged to, as far as actual policies go, then that is a very important thing to know; especially if a non-uFAI existential risk starts to look imminent (but the game theory in that case is crazy). It is not probable but there's a hell of a lot of structural uncertainty and Omohundro's AI drives are still pretty informal. I am still not absolutely sure I know how a self-modifying superintelligence would interpret or reflect on its utility function or terms therein (or how it would reflect on its implicit policy for interpreting or reflecting on utility functions or terms therein). The apparent rigidity of Goedel machines might constitute a disproof in theory (though I'm not sure about that), but when some of the terms are sequences of letters like "makeHumansHappy" or formally manipulable correlated markers of human happiness, then I don't know how the syntax gets turned into semantics (or fails entirely to get turned into semantics, as they case may well be). This implies that the actually-implemented-CDT agent has a single level of abstraction/granularity at like the naive realist physical level at which it's proving things about causal relationships. Like, it can't/shouldn't prove causal relationships at the level of string theory, and yet it's still confident that its actions are causing things despite that structural uncertainty, and yet despite the symmetry it for some reason cannot possibly see how switching a few trans

    Sorry if this has already been addressed. I didn't take the time to read all 300 comments.

    It seems to me that if there were an omniscient Omega, the world would be deterministic, and you wouldn't have free will. You have the illusion of choice, but your choice is already known by Omega. Hence, try (it's futile) to make your illusory choice a one-boxer.

    Personally, I don't believe in determinism or the concept of Omega. This is a nice thought experiment though.

    1Rob Bensinger10y
    How does adding indeterminism help make the problem go away? If Omega only predicts correctly 99% of the time, what gets clarified?

    I don't grasp why this problem seems so hard and convoluted. Of course you have to one-box, if you two-box you'll lose for sure. From my perspective two-boxing is irrational...

    If Omega can flawlessly predict the future, this confirms a deterministic world at the atomic scale. To be a perfect predictor Omega would also need to have a perfect model of my brain at every stage of making my "decision" - thus Omega can see the future and perfectly predict whether or not I'm gonna two-box or not.

    If my brain is wired up in such a way as to choose two-box... (read more)

    To perfectly model your thought processes, it would be enough that your brain activity be deterministic; it doesn't follow that the universe is deterministic. The fact that my computer can model a Nintendo well enough for me to play video games does not imply that a Nintendo is built out of deterministic elementary particles, and a Nintendo emulator that simulated every elementary particle interaction in the Nintendo it was emulating would be ridiculously inefficient.

    I'm kind of surprised at how complicated everyone is making this, because to me the Bayesian answer jumped out as soon as I finished reading your definition of the problem, even before the first "argument" between one and two boxers. And it's about five sentences long:

    Don't choose an amount of money. Choose an expected amount of money--the dollar value multiplied by its probability. One-box gets you >(1,000,000*.99). Two-box gets you <(1,000*1+1,000,000*.01). One-box has superior expected returns. Probability theory doesn't usually encounte... (read more)

    I would take box B, because it would be empty.

    I see your general point, but it seems like the solution to the Omega example is trivial if Omega is assumed to be able to predict accurately most of the time:
    (letting C = Omega predicted correctly; let's assume for simplicity that Omega's fallibility is the same for false positives and false negatives)

    • if you chose just one box, your expected utility is $1M * P(C)
    • if you chose both boxes, your expected utility is $1K + $1M (1 - P(C))
      Setting these equal to find the equilibrium point:
      P(C) = 1000 + 1000000 (1 - P(C))
      P(C) = 1001 - 1000
    ... (read more)

    It certainly seems like a simple resolution exists...

    As a rationalist, there should only ever be one choice you make. It should be the ideal choice. If you are a perfectly rational person, you will only ever make the ideal choice. You are certainly at least, deterministic. If you can make the ideal choice, so can someone else. That means, if someone knows your exact situation (trivial in the Newcomb paradox, as the super intelligent agent is causing your situation) then they can predict exactly what you will do, even without being perfectly rational themse... (read more)

    Welcome to Less Wrong! Why do you think so?
    I think so too. Perhaps we've all heard a slightly different wording of the paradox (or more), but I don't see what causation has to do with it.
    He knows what your environmental circumstances are because he put you in them. That is, he obviously knows that you are going to be encountering a Newcomblike problem because he just gave it to you. (ie. No deep technical meaning, just the obvious.)
    Maybe I'm being dense. Omega needs to know more than just that you are going to encounter the problem, even Omega's scheduler and publicist know that! Omega knows the exact situation, including how an identical model of you would act/has acted, because that is stipulated, but it does not follow trivially from Omega's causing your situation.

    Well, for me there are two possible hypothesis for that :

    1. The boxes are not what they seem. For example, box B contains nano-machinery that detects if you one-box or not, create money if you one-box, and then self-destruct the nano-machinery.

    2. Omega is smart enough to be able to predict if I'll one-box or two-box (he scanned my brain, runned it in a simulation, and saw my I do... I hope he didn't turn off the simulation afterwards, or he would have killed "me" then !).

    In both cases, I should one-box. So I'll one-box. I don't really get the ra... (read more)

    It's strange. I perfectly agree with the argument here about rationality - the rationality I want is the rationality that wins, not the rationality that is more reasonable. This agrees with my privileging truth as a leading which is useful, not which necessarily makes the best predictions. But in other points on the site, it always seems that correspondence is privileged over value.

    As for Newcombs paradox, I suggest writing out all the relevant propositions a la Jaynes, with non-zero probabilities for all propositions. Make it a real problem, not an ideali... (read more)

    An amusing n=3 survey of mathematics undergrads at Trinity Cambridge:

    1) Refused to answer. 2) It depends on how reliable Omega is/but you cant (shouldn't) really quantify ethics anyway/this situation is unreasonable. 3) Obviously 2 box, one boxing is insane.

    3 said he would program an AI to one box. And when I pointed out that his brain was built of quarks just like the AI he responded that in that case free will didn't exist and choice was impossible.

    Upvoted for this sentence:

    "If it ever turns out that Bayes fails - receives systematically lower rewards on some problem, relative to a superior alternative, in virtue of its mere decisions - then Bayes has to go out the window."

    This is such an important concept.

    I will say this declaratively: The correct choice is to take only box two. If you disagree, check your premises.

    "But it is agreed even among causal decision theorists that if you have the power to precommit yourself to take one box, in Newcomb's Problem, then you should do so. If y... (read more)

    "If it ever turns out that Bayes fails - receives systematically lower rewards on some problem, relative to a superior alternative, in virtue of its mere decisions - then Bayes has to go out the window."

    This is such an important concept.

    Yes, but like falsifiability, dangerous. This also goes for 'rationalists win', too.

    'We' (Bayesians) face the Duhem-Quine thesis with a vengeance: we have often found situations where Bayes failed. And then we rescued it (we think) by either coming up with novel theses (TDT) or carefully analyzing the problem or a related problem and saying that is the real answer and so Bayes works after all (Jaynes again and again). Have we corrected ourselves or just added epicycles and special pleading? Should we just have tossed Bayes out the window at that point except in the limited areas we already proved it to be optimal or useful?

    This can't really be answered.

    I liked the quote not because of any notion that Bayes will or should "go out the window," but because, coming from a devout (can I use that word?) Bayesian, it's akin to a mathematician saying that if 2+2 ceases to be 4, that equation goes out the window. I just like what this says about one's epistemology -- we don't claim to know with dogmatic certainty, but in varying degrees of certainty, which, to bring things full circle, is what Bayes seems to be all about (at least to me, a novice). More concisely, I like the quote because it draws a line. We can rail against the crazy strict Empiricism that denies rationality, but we won't hold to a rationality so devoutly that it becomes faith.
    Duhem-Quine is just as much a problem there; from Ludwig Wittgenstein, Remarks on the Foundations of Mathematics: Indeed. To generalize, when we run into skeptical arguments employing the above circularity or fundamental uncertainties, I think of Kripke:

    I think it is important to make a distinction between what our choice is now, while we are here, sitting at a computer screen, unconfronted by Omega, and our choice when actually confronted by Omega. When actually confronted by Omega, your choice has been determined. Take both boxes, take all the money. Right now, sitting in your comfy chair? Take the million-dollar box. In the comfy chair, the contra-factual nature of the experiment basically gives you an Outcome Pump. So take the million-dollar box, because if you take the million-dollar box, it's full of a million dollars. But when it actually happens, the situation is different. You aren't in your comfy chair anymore.

    I'm not in my comfy chair any more, and I still take the million. Why wouldn't I?
    Because the million is already there, along with the thousand. Why not get all of it?
    The million isn't there, because Omega's simulation was of you confronting Omega, not of you sitting in a comfy chair.
    You aren't doublethinking hard enough, then.
    I don't know if this is a joke - I have a poor sense of humour - but you do know Omega predicts your actual behaviour, right? As in, all things taken into account, what you will actually do.
    I am being somewhat ... absurd, and on purpose, at that. But I have enough arrogance lying around in my brain to believe that I can trick the super-intelligence.
    Sorry - I'm always inclined to take people on the internet literally. I used to mess with my friends using the same kind of ow-my-brain Prisoner's-dilemma somersaults, and still I couldn't recognise a joke.
    That's alright. My humor, in real life, is based entirely on the fact that only I know I'm joking at the time, and the other person won't realize it until three days later, when they spontaneously start laughing for no reason they can safely explain. Is that asinine? Yes. Is it hilarious? Hell, yes. So I apologize. I'll try not to do that.
    Not especially, unfortunately. There is something to be said for appearing that you don't give a @#%! whether other people get your humor in real time but it works best if you care a whole lot about making your humor funny to your audience at the time and then just act like you don't care about the response you get. Even if people get your joke three days later you still typically end up slightly worse off for the failed transaction.
    Ah. Wrong referent. It's hilarious for me, and it may, at some point, be hilarious for them. But it's mostly funny for me. That would be why I took time to mention that it was also, in fact, asinine.
    Because I'd end up with only a thousand, as opposed to a million. And I want the million.
    If Omega has already left, I open box B first, take whatever is in it, and then open box A.

    I guess my cognition just breaks down over the idea of Omega. To me, Newcomb's problem seems akin to a theological argument. Either we are talking about a purely theoretical idea that is meant to illustrate abstract decision theory, in which case I don't care how many boxes I take, because it has no bearing on anything tied to reality, or we are actually talking about the real universe, in which case I take both boxes because I don't believe in alien superintelligences capable of foreseeing my choices any more than I believe in an anthropomorphic deity.

    Labeling "I decide to lose" as a snark just seems odd. You are confused. Using Omega is merely a simplification of real possible situations. That is, any situation in which you and the other player have some degree of mutual knowledge. Since those situations are complicated they will sometimes call for cooperation (one boxing, here) but often other considerations or insufficient mutual knowledge will override and call for defection (two boxing). If you wish to consider the effect of just, say, the mass of a cow then assuming a spherical cow in a vacuum is useful. If the conclusion you reach about the mass of said cow doesn't suit you and you say "but there are no spherical cows in vacuums!" then you are using an excuse to avoid biting the bullet, not showing your superior awareness of reality.
    Yeah, that's generally what "I guess my cognition breaks down" means. I think you can reasonably expect people to behave in real life as if they expect the laws of physics to approximate reasonably closely what newtonian mechanics predicts about spherical point masses. What I was saying, however, is that you would be wrong to predict that I defect in prisoners' dilemmas based on my 2-boxing, because for me Newcomb's problem isn't connected to those problems for reasons already stated. I hypothesize that I am not alone in that.
    And I said you are confused regarding this belief and the stated reasons. I don't doubt that others are confused as well - it's a rather common response.

    If in 35 AD you were told that there were only 100 people who had seen Jesus dead and entombed and then had seen him alive afterwards, and that there were no people who had seen him dead and entombed who had seen his dead body afterwards, would you believe he had been resurrected?

    In Newcomb's problem as stated, we are told 100 people have gotten the predicted answer. Then no matter how unlikely our priors put on a superintelligent alien being able to predict what we would do, we should accept this as proof.

    This seems like a pretty symmetric question t... (read more)

    Really? A Phd ? Seriously ?

    If Omega said "You shall only take Box B or I will smite thee" and then proceeded to smite a 100 infidels who dared to two box the rational choice would be obvious (especially if the smiting happened after O left)

    is this really difficult to show mathematicly ?

    This thread has gone a bit cold (are there other ones more active on the same topic?)

    My initial thoughts: if you've never heard of Newcomb's problem, and come across it for the first time in real-time, then as soon as you start thinking about it, the only thing to do is 2-box. Yes, Omega will have worked out you'll do that, and you'll only get $1000, but the contents of the boxes are already set. It's too late to convince Onega that you're going to 1 box.

    On the other hand, if you have already heard and thought about the problem, the rational thing to do is... (read more)

    It seems to me that if I've never before been exposed to Newcomb's problem, and Omega presents me with it, there are two possibilities: either I will one-box, or I will two-box. If I one-box (even without having precommited to doing so, simply by virtue of my thoughts at the moment about the boxes), Omega will have previously worked out that I'm the sort of person who would one-box. Why do you say that the only thing to do in the absence of precommitment is two-box?
    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?
    Well, Newcomb's problem is simple enough that evidential decision theory suffices.
    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).
    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
    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
    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
    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.
    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-
    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.
    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.)
    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.
    See TDT, UDT, ADT.
    Thanks for this... I'm looking at them. If I'm correct, the general thrust seems to be "there is a problem with both causal decision theory and evidential decision theory, since they sometimes recommend different things, and sometimes EDT seems right, whereas at other times CDT seems right. So we need a broader theory". I'm not totally convinced of this need, since I think that in many ways of interpreting the Newcomb problem, EDT and CDT lead to essentially the same conclusion. They both say pre-commit to 1-boxing. If you haven't precommitted, they both say 2-box (in some interpretations) or they both say 1-box (in other interpretations). And the cases where they come apart are metaphysically rather problematic (e.g. Omega's predictions must be perfect or nearly-so without pre-cognition or reverse causation; Omega's simulation of the 2-boxer must be accurate enough to catch the rare occasions when he 1-boxes, but without that simulation itself becoming sentient.) However, again, thanks for the references and for a few new things to think about.
    You seem to be thinking about Omega as if he's a mind-reader that can only be affected by your thoughts at the time he set the boxes, instead of a predictor/simulator/very good guesser of your future thoughts. So it's not "too late". What does it matter if you'll do it reflexively or after a great deal of thought? The problem doesn't say that reflexive decisions are easier for Omega to guess than ones following long deliberation.
    I'm modelling Omega as a predictor whose prediction function is based on the box-chooser's current mental state (and presumably the current state of the chooser's environment). Omega can simulate that state forward into the future and see what happens, but this is still a function of current state. This is different from Omega being a pre-cog who can (somehow) see directly into the future, without any forward simulation etc.
    Yes. And what Omega discovers as a result of performing the simulation depends on what decision you'll make, even if you encounter the problem for the first time, since a physical simulation doesn't care about cognitive novelty. Assuming you're digitally encoded, it's a logically valid statement that if you one-box, then Omega's simulation says that you one-boxed, and if you two-box, then Omega's simulation says that you two-boxed. In this sense you control what's in the box.
    I think this is the disconnect... The chooser's mental state when sampled by Omega causes what goes into the box. The chooser's subsequent decisions don't cause what went into the box, so they don't "control" what goes into the box either. Control is a causal term...
    The goal is to get more money, not necessarily to "causally control" money. I agree that a popular sense of "control" probably doesn't include what I described, but the question of whether that word should include a new sense is a debate about definitions, not about the thought experiment (the disambiguating term around here is "acausal control", though in the normal situations it includes causal control as a special case). So long as we understand that I refer to the fact that it's logically valid that if you one-box, then you get $1,000,000, and if you two-box, then you get only $1,000, there is no need to be concerned with that term. Since it's true that if you two-box, then you only get $1,000, then by two-boxing you guarantee that it's true that you two-box, ergo that you get $1000. Correspondingly, if you one-box, that guarantees that it's true that you get $1,000,000. (The subtlety is hidden in the fact that it might be false that you one-box, in which case it's also true that your one-boxing implies that 18 is a prime. But if you actually one-box, that's not the case! See this post for some discussion of this subtlety and a model that makes the situation somewhat clearer.)

    This is an old thread, but I can't imagine the problem going away anytime soon, so let me throw some chum into the waters;

    Omega says; "I predict you're a one boxer. I can understand that. You've got really good reasons for picking that, and I know you would never change your mind. So I'm going to give you a slightly different version of the problem; I've decided to make both boxes transparent. Oh and by the way, my predictions aren't 100% correct."

    Question: Do you make any different decisions in the transparent box case?
    If so, what was there ... (read more)

    Box B is already empty or already full [and will remain the same after I've picked it]

    Do I have to believe that statement is completely and utterly true for this to be a meaningful exercise? It seems to me that I should treat that as dubious.

    It seems to me that Omega is achieving a high rate of success by some unknown good method. If I believe Omega's method is a hard-to-detect remote-controlled money vaporisation process then clearly I should one-box.

    A super intelligence has many ways to get the results it wants.

    I am inclined to think that I don't kno... (read more)