Newcomb's Problem and Regret of Rationality

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.

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.

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.

Agree/disagree?

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?

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.

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.

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 states into something indeterminate?

If you reject that kind of indeterminacy, what do you imagine happening, if you vacillate and consider taking both boxes? 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.

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?

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 the one with the million dollars.

Since induction applied to the known facts suggests that I can effectively determine the outcome by making a decision, I will select the outcome that I prefer, and choose to take only box B.

Why exactly is that irrational, again?

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 meaningless anyway: you'll do what you'll do, so stop stressing about it.

If b), then you should be a one-boxer for perfectly ordinary rational reasons, namely that it brings it about that you get a million bucks with probability 1.

So there's no problem!

Laura,

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 a), then practical reason is meaningless anyway: you'll do what you'll do, so stop stressing about it.

Deterministic != meaningless. Your action is determined by your motivations, and by your decision process, which may include your stressing about it. It makes perfect sense to say: "My future decision is determined, and my stressing about it is determined; but if-counterfactual I didn't stress about it, then-counterfactual my future decision would be different, so it makes perfect sense for me to stress about this, which is why I am deterministically doing it."

The past can't change - does not even have the illusion of potential change - but that doesn't mean that people who, in the past, committed a crime, are not held responsible just because their action and the crime are now "fixed". It works just the same way for the future. That is: a fixed future should present no more problem for theories of moral responsibility than a fixed past.

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 fact going to take both boxes, and so I will get amount x+1000, namely 1000 more than how much is in fact in the other box.

From this it is obvious that x in the two statements has a different value, and so his conclusion that he will get more if he takes both boxes is false. For x has the value 1,000,000 in the first case, and 0 in the second. He mistakenly assumes it has the same value in the two cases.

Likewise, when the two-boxer says to the one boxer, "If you had taken both boxes, you would have gotten more," his statement is counterfactual and false. For if the one-boxer had been a two boxer, there originally would have been nothing in the other box, and so he would have gotten only $1000 instead of $1,000,000.

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). If the stress IS fixed, then there's no decision left to take. (Except possibly whether or not to stress about the stress, call that stress*, and recurse the argument accordingly.)

In general, for any pair of actions X and Y, where X is determined, either X is conditional on Y, in which case Y must also be determined, or not conditional on Y, in which case Y can be either determined or non-determined. So appealing to Y as part of the process that leads to X doesn't mean that something we could do to Y makes a difference if X is determined.

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 programmer will have known this, so I will get 1,000,000. If I take both boxes, the programmer will have known this, so I will get 1,000. It is better to get 1,000,000 than 1,000. So I choose to take only one box."

If someone tries to confuse the AI by saying, "if you take both, you can't get less," the AI will respond, "I can't take both without different code, and if I had that code, the programmer would have known that and would have put less in the box, so I would get less."

Or in other words: it is quite possible to make a decision, like the AI above, even if everything is fixed. For you do not yet know in what way everything is fixed, so you must make a choice, even though which one you will make is already determined. Or if you found out that your future is completely determined, would you go and jump off a cliff, since this could not happen unless it were inevitable anyway?

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 subjective experiences. To illustrate the distinction using your other example, I could state that I prefer to save 400 lives certainly, simply because the purple fairy in my closet tells me to (my arbitrary preferred objective), and that would be perfectly legitimate. It would only be incoherent if I also declared it to be a strategy which would maximise the number of lives saved if a majority of people adopted it in similar circumstances (a different arbitrary preferred objective). I could in fact have as preferred meta-objective for the universe that all the squilth in flobjuckstooge be globberised, and that would be perfectly legitimate. An FAI (or a BFG, for that matter (Roald Dahl, not Tom Hall)) could scan me and work towards creating the universe in which my proposition is meaningful, and make sure it happens. If now someone else's preferred meta-objective for the universe is ensuring that the princess on page 3 gets a fairy cake, how is the FAI to prioritise ?

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 deciding what to do. So assuming fatalism (which is what we must do if the AI knows what we're going to do, perfectly, in advance) demotivates rationality.

Here's the ultimate problem: our intuitions about these sorts of questions don't work, because they're fundamentally rooted in our self-understanding as agents. It's really, really hard for us to say sensible things about what it might mean to make a "decision" in a deterministic universe, or to understand what that implies. That's why Newcomb's problem is a problem -- because we have normative principles of rationality that make sense only when we assume that it matters whether or not we follow them, and we don't really know what it would mean to matter without causal leverage.

(There's a reason free will is one of Kant's antimonies of reason. I've been meaning to write a post about transcendental arguments and the limits of rationality for a while now... it'll happen one of these days. But in a nutshell... I just don't think our brains work when it comes down to comprehending what a deterministic universe looks like on some level other than just solving equations. And note that this might make evolutionary sense -- a creature who gets the best results through a [determined] causal chain that includes rationality is going to be selected for the beliefs that make it easiest to use rationality, including the belief that it makes a difference.)

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 have to consider the benefits of various possibilities, despite the fact that there is only one possibility, just like there is really only one possibility for the chess program.

So when you considered the various "possibilities", would "jumping off a cliff" evaluate as equal to "going on with life", or would the latter evalulate as better? I suspect you would go on with life, just like a chess program moves its queen to avoid being taken by a pawn, despite the fact that it was totally determined to do this.

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 future and observe your action before it happens. In this case what you do is directly affecting what the box contains, and the problem's statement that whatever you choose won't affect the contents of the box is just wrong.

Or maybe it has a copy of the entire state of your brain, and can simulate you in a software sandbox inside its own mind long enough to see what you will do. In this case it makes sense to think of the box as not being empty or full until you've made your choice, if you are the copy in the sandbox. If you aren't the copy in the sandbox then you'd be better off choosing both boxes, but the way the problem's set up you can't tell this. You can still try to maximize future wealth. My arithmetic says that choosing Box B is the best strategy in this case. (Mixed strategies, where you hope that the sandbox version of yourself will randomly choose Box B alone and the outside one will choose both, are dominated by choosing Box B. Also I assume that if you are in the sandbox, you want to maximize the wealth of the outside agent. I think this is reasonable because it seems like there is nothing else to care about, but perhaps someone will disagree.)

You could interpret Omega differently than in these stories, although I think my first point above that you should think of your choice as causing Omega to put money in the box, or not, is reasonable. I would say that the fact that Omega put the money in the box chronologically before you make the decision is irrelevant. I think uncertainty about an event that has already happened, but that hasn't been revealed to you, is basically the same thing as uncertainty about something that hasn't happened yet, and it should be modeled the same way.

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 question I choose box B. Either I get the million $ -- or I have proven an extragalactical superintelligence wrong. How cool is that? 1000$? Have you looked at the exchange rates lately?

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 observing him being correct 100/100 times assuming a many worlds interpretation of QM). Or he could have just got lucky with the last 100 people he tried it on.

If it is the deterministic option, then what do the counterfactuals about choosing the other box even mean? My approach is to say that 'You could choose X' means that if you had desired to choose X, then you would have. This is a standard way of understanding 'could' in a deterministic universe. Then the answer depends on how we suppose the world to be different to give you counterfactual desires. If we do it with a miracle near the moment of choice (history is the same, but then your desires change non-physically), then you ought two-box as Omega can't have predicted this. If we do it with an earlier miracle, or with a change to the initial conditions of the universe (the Tannsjo interpretation of counterfactuals) then you ought one-box as Omega would have predicted your choice. Thus, if we are understanding Omega as extrapolating your deterministic thinking, then the answer will depend on how we understand the counterfactuals. One-boxers and Two-boxers would be people who interpret the natural counterfactual in the example in different (and equally valid) ways.

If we understand it as Omega using a quantum suicide method, then the objectively right choice depends on his initial probabilities of putting the million in the box. If he does it with a 50% chance, then take just one box. There is a 50% chance the world will end either choice, but this way, in the case where it doesn't, you will have a million rather than a thousand. If, however, he uses a 99% chance of putting nothing in the box, then one-boxing has a 99% chance of destroying the world which dominates the value of the extra money, so instead two-box, take the thousand and live.

If he just got lucky a hundred times, then you are best off two-boxing.

If he time travels, then it depends on the nature of time-travel...

Thus the answer depends on key details not told to us at the outset. Some people accuse all philosophical examples (like the trolley problems) of not giving enough information, but in those cases it is fairly obvious how we are expected to fill in the details. This is not true here. I don't think the Newcomb problem has a single correct answer. The value of it is to show us the different possibilities that could lead to the situation as specified and to see how they give different answers, hopefully illuminating the topic of free-will, counterfactuals and prediction.

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 taking the risk that Omega is not smart enough to have modelled your consciousness sufficiently well. How big is this risk? 100 times out of 100 speaks for itself. Omega is cleverer than we can understand. Box B.

(Time travel? No thanks. I find the probability that Omega is simulating people's minds a hell of a lot more likely than that he's time travelling, destroying the universe etc. And even if he were, Box B!)

If you can have your brain modelled exactly - to the point where there is an identical simulation of your entire conscious mind and what it perceives - then a lot of weird stuff can go on. However, none of it will violate causality. (Quantum effects messing up the simulation or changing the original? I guess if the model could be regularly updated based on the original...but I don't know what I'm talking about now ;) )

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," because it will disintegrate if you do not choose A. I also have no experience with trans-galactic super-intelligences, and its ability to make time-traveling super-boxes is already covered by the discussion above. I think of boxes as things that either are full or are not, independent of my intentions, but I also think of them as things that do not disintegrate based on my intentions.

Taking both is equivalent to just taking A. Restate the problem that way: take A and get $1000 or take B and get $1,000,000. Which would you prefer?

I think the problem becomes more amusing if box A does not disintegrate. They are just two cardboard boxes, one of which is open and visibly has $1000 in it. You don't shout your intention to the sky, you just take whatever boxes you like. The reasonable thing to do is open box B; if it is empty, take box A too; if it is full of money, heck, take box A too. They're boxes, they can't stop you. But that logic makes you a two-boxer, so if Omega anticipates it, and Omega does, B will be empty. You definitely need to pre-commit to taking only B. Assume you have, and you open B, and B has $1,000,000. You win! Now what do you do? A is just sitting there with $1000 in it. You already have your million. You even took it out of the box, in case the box disintegrates. Do you literally walk away from $1000, on the belief that Omega has some hidden trick to retroactively make B empty? The rule was not that the money would go away if you took both, the rule is that B would be empty. B was not empty. A is still there. You already won for being a one-boxer, does anything stop you from being a two-boxer and winning the bonus $1000?

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 to them - then an intertemporal preference reversal starts looking like just another preference reversal.

*) I developed my decision theory using mathematical technology, like Pearl's causal graphs, that wasn't around when causal decision theory was invented. (CDT takes counterfactual distributions as fixed givens, but I have to compute them from observation somehow.) So it's not surprising if I think I can do better.

*) We're not talking about a patchwork of self-modifications. An AI can easily generally modify its future self once-and-for-all to do what its past self would have wished on future problems even if the past self did not explicitly consider them. Why would I bother to consider the general framework of classical causal decision theory when I don't expect the AI to work inside that general framework for longer than 50 milliseconds?

*) I did work out what an initially causal-decision-theorist AI would modify itself to, if it booted up on July 11, 2018, and it looks something like this: "Behave like a nonclassical-decision-theorist if you are confronting a Newcomblike problem that was determined by 'causally' interacting with you after July 11, 2018, and otherwise behave like a classical causal decision theorist." 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 others.

*) Imagine time spread out before you like a 4D crystal. Now imagine pointing to one point in that crystal, and saying, "The rational decision given information X, and utility function Y, is A", then pointing to another point in the crystal and saying "The rational decision given information X, and utility function Y, is B". Of course you have to be careful that all conditions really are exactly identical - the agent has not learned anything over the course of time that changes X, the agent is not selfish with temporal deixis which changes Y. But if all these conditions are fulfilled, I don't see why an intertemporal inconsistency should be any less disturbing than an interspatial inconsistency. You can't have 2 + 2 = 4 in Dallas and 2 + 2 = 3 in Minneapolis.

*) What happens if I want to use a computation distributed over a large enough volume that there are lightspeed delays and no objective space of simultaneity? Do the pieces of the program start fighting each other?

*) Classical causal decision theory is just not optimized for the purpose I need a decision theory for, any more than a toaster is likely to work well as a lawnmower. They did not have my design requirements in mind.

*) I don't have to put up with dynamic inconsistencies. Why should I?

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 box depends on what choice we make. The fact that the placement of money into the box happened in the past is irrelevant, because we've already presumed that the relevant cause-and-effect relationship works backwards in time.

Eliezer states that the past is fixed. Well, it may be fixed in some absolute sense (although that is a complicated topic), but from our ignorant perspective we have to consider what appears to us to be the possible alternatives. That means that we must consider the money in the boxes to be uncertain. Choosing causes Omega to put a particular amount of money in the box. That this happened in the past is irrelevant, because the causal dependence points into the past instead of the future.

Even if we ignore actual time travel, we must consider the amount of money present to be uncertain until we choose, which then determines how much is there - in the sense of our technique, from our limited perspective.

If we accept that Omega is really as accurate as it appears to be - which is not a small thing to accept, certainly - and we want to maximize money, then the correct choice is B.

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 others.

This is a genuine concern. Note that most instances of precommitment arise quite naturally due to reputational concerns: any agent which is complex enough to come up with the concept of reputation will make superficially irrational ("hawkish") choices in order not to be pushed around in the future. Moreover, precommitment is only worthwhile if it can be accurately assessed by the counterparty: an agent will not want to "generally modify its future self ... to do what its past self would have wished" unless it can gain a reputational advantage by doing so.

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 utility lets us add up conflicting incentives and reduce to a single preference: a multiplicity of strategic incentives is not an excuse for inconsistency.

I am a Bayesian; I don't believe in probability calculations that come out different ways when you do them using different valid derivations. Why should I believe in decisional calculations that come out in different ways at different times?

I'm not sure that even a causal decision theorist would agree with you about strategic inconsistency being okay - 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, because in the former case you cause Omega's action while in the latter case you do not. In other words, they would insist the two situations are importantly different, not that time inconsistency is okay.

And I observe again that a self-modifying AI which finds itself with time-inconsistent preferences, strategies, what-have-you, will not stay in this situation for long - it's not a world I can live in, professionally speaking.

Trying to find a set of preferences that avoids all strategic conflicts between your different actions seems a fool's errand.

I guess I completed the fool's errand, then...

Do you at least agree that self-modifying AIs tend not to contain time-inconsistent strategies for very long?

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 expectation would be $990,000 and a two-boxer would have an expectation of $11,000.

But the whole argument seems to be about what extra knowledge you have; in particular, Can causation work in reverse? or Is Omega really superintelligent? or even Are the conditions stated in the problem logically inconsistent (which would justify any answer.)

Perhaps someone who enjoys these kinds of odds calculations could investigate the extent to which we know these things and how it affects the outcome?

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 500 years and then die, with certainty.
B: Live forever, with probability 0.000000001%; die within the next ten seconds, with probability 99.999999999%

Do you choose A or B? Is it possible to choose A and have an unbounded utility function with respect to life? It seems to me that an unbounded utility function implies the choice of B. But then what if the probability of living forever becomes one in a googleplex, or whatever? Of course, this is a kind of Pascal's Wager; but it seems to me that your utility function implies that you should accept the Wager.

It also seems to me that the intuitions suggesting to you and others that Pascal's Mugging should be rejected similarly are based on an intuition of a bounded utility function. Emotions can't react infinitely to anything; as one commenter put it, "I can only feel so much horror." So to the degree that people's preferences reflect their emotions, they have bounded utility functions. In the abstract, not emotionally but mentally, it is possible to have an unbounded function. But if you do, and act on it, others will think you a fanatic. For a fanatic cares infinitely for what he perceives to be an infinite good, whereas normal people do not care infinitely about anything.

This isn't necessarily against an unbounded function; I'm simply trying to draw out the implications.

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. Your other option is to pay nothing for a chance to win B if Omega is wrong.

Then just make your bet based on what you know about Omega. As stated, we only know his track record over 100 attempts, so use that. Don't worry about the nature of causality or whether he might be scanning your brain. We don't know those things.

If you do it that way, you'll probably find that your answer depends on A and B as well as Omega's track record.

I'd probably put Omega at around 99%, as Hendrik did. Keeping A at a thousand dollars, I'd one-box if B were a million dollars or if B were something I needed to save my life. But I'd two-box if B were a thousand dollars and one cent.

So I think changing A and B and declaring that your strategy must stay the same is invalid.

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 others.

If your choice has some %age chance of causing Omega's action, then the problem becomes one of risk management. What is your chance of getting the big award if you choose B compared with the utility of the two chocies.

I agree with what Tom posted. The only paradox here is that the problem both states that your choice causes Omega's action (because it supposedly predicts perfectly), and also says that your action does not cause Omega's action (because the decision is already made). Thus, wether or not you think box B, or both boxes is the correct choice, depends on which of these two contradictory statements you end up believing.

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 me and extrapolate Omega's extrapolated decision. If there is a million in box B I pick both otherwise just B.

Looks like Omega will be wrong either way, or will I be wrong? Or will the universe crash?

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 winning $1000000.) So rationally, stay at home and hope your candidate wins, right? And then you'll have avoided the inconvenience too. Take both boxes.

But here's the twist. If you muster the will to vote, it stands to reason that those of a similar mind to you (a potentially statistically significant number of people) would also muster the will to vote, because of their similarity to you. So knowing this, why not stay home anyway, avoid the inconvenience, and trust all those others to vote and win the election? They're going to do what they're going to do. Your actions can't change that. The contents of the boxes can't be changed by your actions. Well, if you don't vote, perhaps that means neither will the others, and so it goes. Therein lies the similarity to Newcomb's problem.

"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 usual laws of physics.

Once we've done enough experiments that we're forced into the hypothesis that the boxes are an exception to the usual laws of physics, it's pretty clear what to do. The obvious revised laws of physics based on the new observations make it clear that one should choose just one box.

So a rational agent would do the right thing, but only because there's no way to get it to believe the backstory.

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 which has complexity K(X + "put $1 million in box B if X does not take box A"), which is slightly greater than K(X).

In the framework of the ideal rational agent in AIXI, the agent guesses that Omega is the shortest program consistent with the observed interaction so far. But it can never guess Omega because its complexity is greater than that of the agent. Since AIXI is optimal, no other agent can make a rational choice either.

As an aside, this is also a wonderful demonstration of the illusion of free will.

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?

OR

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

OR

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 that in any way affects Omega's prediction. For the Irene/Rachel situation, there is no way to ever "precommit;" the subject never gets to play Omega's game again and Omega scans their brains before they ever heard of him. (So imagine you only had one shot at playing Omega's game, and Omega made its prediction before you ever came to this website or anywhere else and heard about Newcomb's paradox. Then that already decides what it puts in the boxes.)

Secondly, I think a requirement of the problem is that your choice, at the time of actually taking the box(es), cannot effect what's in the box. What we have here are two completely different problems; if in any way Omega or your choice information can travel back in time to change the contents of the box, the choice is trivial. So yes, Omega may have chosen to discriminate against rational people and award irrational ones; the point is, there is absolutely nothing we can do about it (neither in precommitment or at the actual time to choose).

To clarify why I think two-boxing is the right choice, I would propose a real life experiment. Let's say we developed a survey, which, by asking people various questions about logic or the paranormal etc..., we use to classify them into one-boxers or two-boxers. The crux of the setup is, all the volunteers we take have never heard of the Newcomb Paradox; we make up any reason we want for them to take the survey. THEN, having already placed money or no money in box B, we give them the story about Omega and let them make the choice. Hypothetically, our survey could be 100% accurate; even if not it may be very accurate such that many of our predicted one-boxers will be glad to find their choice rewarded. In essence, they cannot "precommit" and their choice won't magically change the contents of the box (based on a human survey). They also cannot go back and convince themselves to cheat on our survey - it's impossible - and that is how Omega is supposed to operate. The point is, from the experimental point of view, every single person would make more from taking both boxes, because at the time of choice there's always the extra $1000 in box A.

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 deterministic process that feels like a free choice, and you will find the million. If the latter, you will have the subjective experience of "deciding to take both boxes", and you will find nothing in the opaque box.

In short, the framing of the problem implies that your decision-making process is deterministic (which does not preclude it being a process that you are conscious of participating in), and the figurative notion of "free will" does not include literal degrees of freedom. If you must insist on viewing it as a question of what the correct action is, it's to take the one box. Regardless of your motivation, even if your reason for doing so is this argument, you will find yourself in a universe in which events (including thought events) have led you to take one box, and these are the same universes in which the alien places a million dollars in the box.

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.

If you have a solution to Newcomb's Problem, but don't have the time to work on it, is there any chance you will post a sketch of your solution for other people to investigate and/or develop?

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 "preferring 50 Years of Torture over 3^^^3 Dust Specks" as an unmitigated loss. A moral theory that returns a result that almost every moral person alive would view as abhorrent has at least one flaw if it could produce such a major loss.)

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 brain scan. I don't see anything about this scenario which is absurd, impossible, or even particularly low probability. The one problem is that there might well be a certain fraction of people for whom you really can't predict what they'll do, because they're right on the edge and will decide more or less at random. But you could exclude them from the experiment and just give those with solid predictions a shot at the boxes.

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 sure that what Eliezer has in mind is,

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

This indeed means that {u(live n' years) | n' in N} is not upwards bounded -- I did check the math :-) --, which means that u is not upwards bounded, which means that u is not bounded. But what he actually said was,

(B) For all n, (p+q)u(live n years) <= pu(live forever) + q*u(live googolplex years)

That's not only different from A, it contradicts A! It doesn't imply that u needs to be bounded, of course, but it flat out states that {u(live n years) | n in N} is upwards bounded by (pu(live forever) + qu(live googolplex years))/(p+q).

(We may perhaps see this as reason enough to extend the domain of our utility function to some superset of the real numbers. In that case it's no longer necessary for the utility function to be unbounded to satisfy (A), though -- although we might invent a new condition like "not bounded by a real number.")

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 conclude that u(live forever) = the limit of u(live n years) cannot be finite. Okay. Without the qualification I was missing, the condition wouldn't be inconsistent with a bounded utility function, since the difference wouldn't have to get arbitrarily small, but the qualification certainly seems reasonable.

(I would still prefer for all possibilities considered to have defined utilities, which would mean extending the range of the utility function beyond the real numbers, which would mean that u(live forever) would, technically, be an upper bound for {u(live n years) | n in N} -- that's what I had in mind in my last paragraph above. But you're not required to share my preferences on framing the issue, of course :-))

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, not meant to be solved, but rather explored in different directions. In its original form, much like the Monty Hall problem, Newcomb's scenario is not well stated to give rise to problem with a calculated solution.

This is not a criticism of the problem, indeed it is an ingenious little puzzle.

And there is much to learn from well defined Newcomb like problems.

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 that you are choosing one box is the best way for you to ensure that will be what is predicted.

The standard attack on this line of reasoning seems to be that since his prediction happened in the past, your decision can't influence it. But it already has influenced it. He was aware of the decision before you made it (evidenced by his predictive power). In fact, it is not really a decision in the sense of "freely" choosing one of two options (in the way that most people use "freely" at least). Think of this decision as just extremely complicated and seemingly unpredictable data analysis, where the unpredictability comes from never being able to know intimately every part of the decision process and the inputs. But if one could perfectly crack the "black box" of your decision, as this alien appears to have done (at least this seems by far the most plausible explanation to me) then one could predict decisions with the accuracy the alien possesses. In other words, the gears were already in motion for your decision to be made, and the alien was already witness whether you realized it or not. In that sense you aren't making your decision afterwords when you think you are, you are actually realizing the decision that you were already set up to make at an earlier time.

If you agree with what I have written above, your obvious best decision is to just go ahead and pick one box, and hope that the alien would have predicted this. Based on the evidence, that will probably be enough to make the one million show up. Deciding instead to go for two boxes for any reason whatsoever will probably mean that the million won't be there. The time issue is just an illusion caused by your imperfect knowledge and data processing that takes time.

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.

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 to have a chat with. That chat would be likely to have tremendously more utility for me than $1,000,000.

Is that a valid choice? Does it disprove Omega's infallibility? Is it a rational choice?

If messing with the question is not a constructive addition to the debate, accept my apologies, and flame me lightly, please.

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 in the other

b) with probability 1-p there is 0 in one box, and 1k in the other So:

the expected return of two-boxing is p(1M+1k)+(1-p)1k = 1Mp + 1kp + 1k - 1kp = 1Mp + 1k

the expected return of one-boxing is 1Mp

If the act of choosing affects the prior determination p, then the expected return calculation differs depending on my choice:

If I choose to two-box, then p=~0, and I get about 1k on average

If I choose to one-box, then p=~1, and I get about 1M on average

In this case, the expected return is higher by one-boxing.

If choosing the box does not affect p, then p is the same in both expected return calculations. In this case, two boxing clearly has better expected return than one-boxing.

Of course if the determination of p is effected by the choice actually made in the future, you have a situation with reverse-time causality.

If I know that I am going to encounter this kind of problem, and it is somehow possible to pre-commit to one boxing before the alien determines the probability p of me doing so, that certainly makes sense. But it is difficult to see why I would maintain that commitment when the choice actually presents itself, unless I actually believe this choice effects p, which, again, implies reverse-time causality.

It seems the problem has been setup in a deliberately confusing manner. It is as if the alien has just decided to find people who are irrational and pay them 1M for it. The problem seems to encourage irrational thinking, maybe because we want to believe that rational people always win, when of course one can set up a fairly absurd situation so that they do not.

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?

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 offer Hannibal Lecter the same deal, he commits murder, and then opens the box and finds $1 million. Amazing? I don't think so. I was simply able to create an accurate psychological profile of the two of you.

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 where (quantum interactions and chaos theory aside) all Omega-chosen humans have turned out to be correctly predictable in this one aspect.

Perhaps Omega is deliberately NOT chosing to test humans it can't predict. Or it is able to affect the local spacetime sufficiently to 'lock in' a choice even after it's physically left the area?

We can't tell. It's superintelligent. It's not playing on our field. It's potentially an external source of metalogic. The rules go out the window.

In short, the problem as described is not sufficiently constrained to presume a paradox, because it's not confining itself to a single logic system. It's like asking someone only familiar with non-imaginary numbers what the square root of negative one is. Just because they can't derive an answer doesn't mean you don't have one - you're using different number fields.

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.

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. Presumably causal decision theorists would agree.

However, in any version of the problem where there is no clear evidence of violations of currently known physics and where the money has been placed by Omega before my decisions, I am a two-boxer. Yet I think that your post above must not be talking about the same problem that I am thinking of, especially at the end. Although you never said so, it seems to me that you must be talking about a problem which says "If you choose Box B, then it will have a million dollars; if you choose both boxes, then Box B will be empty.". But that is simply not what the facts will be if Omega has made the decision in the past and currently understood physics applies. In the problem as stated, Omega may make mistakes in the future, and that makes all the difference.

It's presumptuous of me to assume that you're talking about a different problem from the one that you stated, I know. But as I read the psychological states that you suggest that I might have —that I might wish that I considered one-boxing rational, for example—, they seem utterly insane. Why would I wish such a thing? What does it have to do with anything? The only thing that I can wish for is that Omega has predicted that I will be a one-boxer, which has nothing to do with what I consider rational now.

The quotation from Joyce explains it well, up until the end, where poor phrasing may have confused you. The last sentence should read:

When Rachel wishes she was Irene's type she is wishing for Irene's circumstances, not wishing to make Irene's choice.

It is simply not true that Rachel envies Irene's choice. Rachel envies Irene's situation, the situation where there is a million dollars in Box B. And if Rachel were in that situation, then she would still take both boxes! (At least if I understand Joyce correctly.)

Possibly one thing that distinguishes me from one-boxers, and maybe even most two-boxers, is that I understand fundamental physics rather thoroughly and my prior has a very strong presumption against backwards causality. The mere fact that Omega has made successful predictions about Newcomb's Paradox will never be enough to overrule that. Even being superintelligent and coming from another galaxy is not enough, although things change if Omega (known to be superintelligent and honest) claims to be a time-traveller. Perhaps for some one-boxers, and even for some irrational two-boxers, Omega's past success at prediction is good evidence for backwards causality, but not for me.

So suppose that somebody puts two boxes down before me, presents convincing evidence for the situation as you stated it above (but no more), and goes away. Then I will simply take all of the money that this person has given me: both boxes. Before I open them, I will hope that they predicted that I will choose only one. After I open them, if I find Box B empty, then I will wish that they had predicted that I would choose only one. But I will not wish that I had chosen only one. And I certainly will not hope, beforehand, that I will choose only one and yet nevertheless choose two; that would indeed be irrational!

E1 = E($1 000 000) * X  
E2 = E($1 000) + E($1 000 000) * Y

X > Y + E($1 000) / E($1 000 000)

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 people make different decisions and think they are being rational - they are!

If A > B they choose one box, if B > A they choose both boxes.

This also shows why people will change their views if the amount in the visible box is changed (to $990,000 or $10).

Basically, in this instance, if you think the probability of Omega being able to determine your future action is greater than 0.5005 then you select a single box, if less than that you select both boxes. At P(Om)=0.5005 the expected return of both strategies is $500,500.

EDIT. I think I oversimplified B, but the point still stands. nhamann - I didn't see your post before writing mine. I think the only difference between them is that I state that it is a personal view of the probability of Omega being able to predict choices and you seem to want to use the actual probability that he can.

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. Alternatively, we could decide that our supposed evidence is untrustworthy and that we are being lied to when we are told that Omega made 100 successful predictions – we might think that this problem describes a nonsensical, impossible situation, similarly to if we were told that there was a barber who shaves everyone who does not shave themself.

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?

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 those who have never heard of a particular brand of decision theory.)

It's like with file compression. In bitmaps, there are frequently large areas with similar colour. With this fact we can design a system that writes that taking less space. However, if we then try to compress a random bitmap, it will take more space than before the compression. Same thing with human minds. They work simply and relatively efficiently, but there's a whole field dedicated to finding flaws in its method. If you use causal decision theory, you sacrifice your ability at games against superhuman creatures that can predict the future, in return for better decision making when that isn't the case. That seems like a reasonably fair trade-off to me. Any theory which gets this one right opens itself to either getting another one wrong, or being more complex and thus harder for a human to use correctly.

1. The scientific method and what I know of rationality make the initial assumption that your belief does not affect how the world works. "If a phenomenon feels mysterious, that is a fact about our state of knowledge, not a fact about the phenomenon itself." etc. However, this isn't something which we can actually know.

Some Christians believe that if you pray over someone with faith, they will be immediately healed. If that is true, rationalists are at a disadvantage, because they aren't as good at self delusion or doublethink as the untrained. They might never end up finding out that truth. I know that religion is the mind killer too, I'm just using the most common example of the supremely effective standard method being unable to deal with an idea. It's necessarily incomplete.

1. I don't agree with you that "reason" means "choosing what ends up with the most reward". You're mixing up means and end. Arguing against a method of decision making because it comes up with the wrong answer to a specific case is like complaining that mp3 compression does a lousy job of compressing silence. I don't think that reason can be the only tool used, just one of them

Incidentally, I would totally only take the $1000 box, and claim that Omega told me I had won immortality, to confuse all decision theorists involved.

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 this particular situation.

Rules, almost by definition, are inconvenient in some situations. Even a rule that's good on balance, a rule you wouldn't want to discard, will sometimes have negative consequences. Otherwise there would be no need to make it a rule! "Don't fool yourself into believing falsehoods" is a good rule. In some situations it may hurt you, when a delusion might have been happier. The hurt is real, even if it's outbalanced in the long run and in expected value. The regret is real. It's just local.

"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 any of these results in a different game (either one that depends on how Omega handles random strategies, or one which depends on how often Omega is wrong - and we lack information on either)

Second, I'm going to reframe the problem a bit: Omega comes to you and has you write a decision-making function. He will evaluate the function, and populate Box B according to his conclusions on what the function will result in. The function can be self-modifying, but must complete in finite time. You are bound to the decision made by the actual execution of this function.

I can't think of any argument as to why this reframing would produce different results, given both Assumptions A and B as true. I feel this is a valid reframing because, if we assume Omega is in fact infallible, I don't see this as being any different from him evaluating the "actual" decision making function that you would use in the situation. Certainly, you're making a decision that can be expressed logically, and presumably you have the ability to think about the problem and modify your decision based on that contemplation (i.e. you have a decision-making function, and it can be self-modifying). If your decision function is somehow impossible to render mathematically, then I'd argue that Assumption A has been violated and we are, once again, playing a different game. If your decision function doesn't halt in finite time, then your payoff is guaranteed to be $0, since you will never actually take either box >.>

Given this situation, the AI simply needs to do two things: Identify that the problem is Newcombian and then identify some function X that produces the maximum expected payoff.

Identifying the problem as Newcombian should be trivial, since "awareness that this is a Newcombian problem" is a requirement of it being a Newcombian problem (if Omega didn't tell you what was in the boxes, it would be a different game, neh?)

Identifying the function X is well beyond my programming ability, but I will assert definitively that there is no function that produces a highe expected payoff than f(Always One-Box). If I am proven wrong, I dare say the person writing that proof will probably be able to cash in to a rather significant payoff :)

Keep in mind that the decision function can self-modify, but Omega can also predict this. The function "commit to One-Box until Omega leaves, then switch to Two-Box because it'll produce a higher gain now that Omega has made his prediction" would, obviously, have Omega conclude you'll be Two-Boxing and leave you with $0.

I honestly cannot find anything about this that would be overly difficult to program, assuming you already had an AI that could handle game theory problems (I'm assuming said AI is very, very difficult, and is certainly beyond my ability).

Given this reframing, f(Always One-Box) seems like a fairly trivial solution, and neither paradoxical nor terribly difficult to represent mathematically... I'm going to assume I'm missing something, since this doesn't seem to be the concensus conclusion at all, but since neither me nor my friend can figure out any faults, I'll go ahead and make this my first post on LessWrong and hope that it's not buried in obscurity due to this being a 2 year old thread :)

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 situations where we make the wisest decision possible based on available information and it turns out to be wrong; the universe is not fair, we know this already. So I will assume we have all the relevant information needed to win. We will also assume that Omega does have the capability to accurately predict my actions; and that causality is not violated (rationality cannot be expected to win if causality is violated!).

Assuming this, I can have a conversation with Omega before it leaves. Mind you, it's not a real conversation, but having sufficient information about the problem means I can simulate its part of the conversation even if Omega itself refuses to participate and/or there isn't enough time for such a conversation to take place. So it goes like this...

Me: "I do want to gain as much as possible in this problem. For that effect I will want you to put as much money in the box as possible. How do I do that?"

Omega: "I will put 1M$ in the box if you take only it; and nothing if you take both."

Me: "Ah, but we're not violating causality here, are we? That would be cheating!"

Omega: "True, causality is not violated. To rephrase, my decision on how much money to put in the box will depend on my prediction of what you will do. Since I have this capacity, we can consider these synonymous."

Me: "Suppose I'm not convinced that they are truly synonymous. All right then. I intend to take only the one box".

Omega: "Remember that I have the capability to predict your actions. As such I know if you are sincere or not."

Me: "You got me. Alright, I'll convince myself really hard to take only the one box."

Omega: "Though you are sincere now, in the future you will reconsider this decision. As such, I will still place nothing in the box."

Me: "And you are predicting all this from my current state, right? After all, this is one of the parameters in the problem - that after you've placed money in the boxes, you are gone and can't come back to change it".

Omega: "That is correct; I am predicting a future state from information on your current state".

Me: "Aha! That means I do have a choice here, even before you have left. If I change my state so that I am unable or unwilling to two-box once you've left, then your prediction of my future "decision" will be different. In effect, I will be hardwired to one-box. And since I still want to retain my rationality, I will make sure that this hardwiring is strictly temporary."

fiddling with my own brain a bit

Omega: "I have now determined that you are unwilling to take both boxes. As such, I will put the 1,000,000$ in the box."

Omega departs

I walk unthinkingly toward the boxes and take just the one

Voila. Victory is achieved.

My main conclusion is here is that any decision theory that does not allow for changing strategies is a poor decision theory indeed. This IS essentially the Friendly AI problem: You can rationally one-box, but you need to have access to your own source code in order to do so. Not having that would so inflexible as to be the equivalent of an Iterative Prisoner's Dilemma program that can only defect or only cooperate; that is, a very bad one.

The reason this is not obvious is that the way the problem is phrased is misleading. Omega supposedly leaves "before you make your choice", but in fact there is not a single choice here (one-box or two-box). Rather, there are two decisions to be made, if you can modify your own thinking process:

1. Whether or not to have the ability and inclination to make decision #2 "rationally" once Omega has left, and
2. Whether to one-box or two-box.

...Where decision #1 can and should be made prior to Omega's leaving, and obviously DOES influence what's in the box. Decision #2 does not influence what's in the box, but the state in which I approach that decision does. This is very confusing initially.

Now, I don't really know CDT too well, but it seems to me that presented as these two decisions, even it would be able to correctly one-box on Newcomb's problem. Am I wrong?

Eliezer - if you are still reading these comments so long after the article was published - I don't think it's an inconsistency in the AI's decision making if the AI's decision making is influenced by its internal state. In fact I expect that to be the case. What am I missing here?

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 they’re risk-neutral, but they are bound by their beliefs.

1. Two-boxers who defend their decision do so on ground of “no backwards causality” (uh, what’s the smart-people term for that?). They don’t believe in Omega’s omniscience. There’s Q1 of these.

2. Two-boxers who regret their decision also concede to Omega’s near-perfect omniscience. There’re Q2 of these.

3. One-boxers who're happy also concede to Omega’s near-perfect omniscience. There’re Q3 of these.

4. One-boxers who regret foregoing $1000. They don’t believe in Omega’s omniscience. There’re Q4 of these.

I’ll offer groups 2 and 3 (believers in that I’ll only get 1000) to split my 1000 between them, in proportion to their bet, if they’re right. If they believe in Omega’s perfect predictive powers, they think there’s a 0% chance of me winning. Therefore, it’s a good bet for them. Expected profit = 1000/weight-0*(all their money)>0

Groups 1 and 4 are trickier. They think Omega has a P chance of being wrong about me. I’ll ask them to bet X=1001000P/((1-P)weight)-eps, where weight is a positive number >1 that’s a function of how many people donated how much. Explicitly defining weight(Q1, Q4, various money caps) is a medium-difficulty exercise for a beginning calculus student. If you insist, I’ll model it, but it will take me more time than I’d already spent on this. So, for a person in one of these groups, expected profit = -X(1-P)+1001000P/weight = eps > 0!

So what do I have now? (Should I pray to Bayes that my intuition be confirmed?) There’re two possible outcomes of taking both boxes.

1. Both are full. I give the 1001000 to groups 1 and 4, and collect Q21000+Q31000000 from groups 2 and 3, which is more than 1001000 if Q3>0 AND Q2>0, or if Q3>1. This outcome has potential for tremendous profit. Call this number PIE >> 1001000.

2. Only A is full. I split my 1000 between groups 2 and 3, and collect X1Q1+X4Q4 from groups 1 and 4. What are X1 and X4 again? X, the amount of money group 1 and group 4 bet, is unique for each group. I called group 1’s X X1, group 4’s X4.

I need to find the conditions when X1Q1+X4Q4 > 1000. So suppose I undermaximized my profit, and completely ignored the poor group 1 (their 1000 won’t make much difference either way). Then X=X4 becomes much simpler, X=1001000P/((1-P)Q4)-eps, and then they payoff I get is -Q4eps+1001000P/(1-P). P = 0.001 and Q4eps < $2 guarantee X1Q1+X4Q4 > X4Q4 > 1000.

That’s all well and good, but if P is low (under 0.5), I’m getting less than 1001000. What can I do? Hedge again! I would actually go to people of groups 1 and 4 again, except it’s getting too confusing, so let’s introduce a “bank” that has the same mentality as the people of groups 1 and 4 (that there’s a chance P that Omega will be wrong about me). Remember PIE? The bank estimates my chances of getting PIE at P. Let’s say if I don’t get PIE, I get 1000 (which is the lowest possible profit for outcome 2; otherwise it’s not worth making that bet). I ask the following sum from the bank: PIEP+1000(1-P) – eps. The bank makes a profit of eps > 0. Since PIE is a large number, my profit at the end is approximately PIEP+1000(1-P) > 1001000.

Note that I’d been trying to find the LOWER bound on this gambit. Actually plugging in numbers for P and Q’s easily yielded profits in the 5 mil to 50 mil range.

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 silly argument for one-boxing: Omega plausibly makes his decision by running a simulation of you. If you are the real copy, it might be best to two-box, but if you are the simulation then one-boxing earns real-you $1000000. Since you can't distinguish whether this is real-you or simulation-you, you should one-box.

3) Would it change things for people if instead of $1000000 vs $1000 it were $1001 vs $1000? Where is the line drawn?

4) Eliezer: just curious about how you deal with paradoxes about infinity in your utility function. If for each n, on day n you are offered to sacrifice one unit of utility that day to gain one unit of utility on day 2n and one unit on day 2n+1 what do you do? Each time you do it you seem to gain a unit of utility, but if you do it every day you end up worse than you started.

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 punished for my decision to two-box, I am being punished for my test answers, and in reality am probably being punished for having personality traits that correlate well with being a two-boxer. I can rationally regret having the wrong personality traits.

Omega #2: You are sent through the Newcomb dilemma, given an amnesia pill and then sent through for real. Omega's prediction is whatever you did the first time (this is similar to the simulation case). If I know this is going on, I clearly one-box because I don't know whether this is the first time through or the second time through.

Omega #3: Omega makes his prediction by observing me and using a time machine. Clearly I one-box.

Omega #4: It is inscribed in the laws of physics somewhere the Omega cannot make a prediction that comes out wrong. Clearly I one-box.

But I think that the problem as stated is ill posed since I don't know what my probability distribution over Omegas should be (given that it depends a lot on exactly what evidence convinces me that Omega is actually a good predictor).

"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-commitment", or create a new 'decision theory'.

The chain of physical causality just has a branch:

M0-> O(D)-> B

M0-> M1-> M2-> .. MN ->D

and O(D) = D

Where M0, M1, M2 .. . MN are the agent's mind states, D is the agent's decision, O is Omega's prediction of the decision, and B is the content of box B.

Your decision does not physically cause the contents of box B to change. Your decision itself however is caused by your past state of mind, and this prior state is also the cause of the box's current contents (via the power of Omega's predictor). So your decision and the box's contents are casually linked, entangled if you will.

From your perspective, the box's contents are unknown. Your final decision is also unknown to you, undecided, until the moment you make that decision by opening the box. Making the decision itself reveals this information about your mind history to you, along with the contents of the box.

One way of thinking about it is that this problem is an illustration of the dictum that any mind or computational system can never fully predict itself from within.

Note that in the context of actual AI in computer science, this type of reflective search (considering a potential decision, then agent B's consequent decision, your next decision, and so on, exploring a decision tree) is pretty basic stuff. In this case the Omega agent essentially has an infinite branching depth, but the decision at each point is pretty simple - because Omega always gets the 'last move'.

You may start as a 'one boxer', thinking that after the scan, you can now outwit Omega by 'self-modifying' into a 'two-boxer' (which really can be just as simple as changing your internal register), but Omega already predicted this move .. and your next reactive move of flipping back to a 'one-boxer' . . and the next, on and on to infinity . . .until you finally run out of time and the register is sampled. You can continue chaining M's to infinity, but you can't change the fact that MN->D and O(D) = D.

Part of the confusion experienced by the causal decision camp may stem from the subjectivity of the solution.

The optimal decision for some abstract algorithm, divorced from Omega's predictive brainscan, will of course choose to two-box, simply because it's decision is not causally linked to the box's contents.

But your N-box register is linked to the box's contents, so you should set it to 1.

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 formulation he simply declared that the box won't be filled based on the actual decision. Fine. How would he go about proving that to someone actually faced with the scenario? Rational people do not risk a million dollars based on an unprovable statement by a philosopher. Same with claims that, for example, Omega didn't set up the boxes so that two-boxing actually results in the annihilation of the contents of box B. Or that Omega doesn't teleport the money in B somehow after the decider makes the decision to one-box. Those declarations may have a truth value of 1 for purposes of a person outside observing the scenario, but unless empirically testable within the scenario, cannot be valued as approximating 1 by the person making the decision.

Every "given" that the decision-maker can't verify is a "given" that is not usable for making the decision. The whole argument for two-boxing depends on a boundary violation; that the knowledge known by the reader but which cannot be known to the character in the scenario can somehow be used by the character in the scenario to make a decision.

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 be completely obvious and unambiguous.

But since this community has spent a while debating this, I conclude that there's a good chance I have missed something important. What is it?

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 decision.

[1] ie something you didn't think of that makes Box B empty even after Omega's gone away, or an invisible portkey in box B that is activated the moment you pick up Box A, or Omega's time-machine that let him go forward to see your decision before putting the money into the boxes... or a device using some hand-wavey quantum state that lets either Box A be taken or Box B's contents to exist...

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 "to simulate" - i.e. Omega can predict me by running a simulation of me (say using a universal quantum Turing machine) though that is not necessarily the only way to do so. Given that we're in a quantum world, would Omega actually need to simulate me in order to ensure a correlation between its variable and my choice, potentially in another galaxy, of whether to pick A+B or B?

Say |Oab> and |Ob> are the two eigenstates of Omega's variable (w.r.t. some operator it has) and the box system in front of me similarly has two eigenstates |Cab> and |Cb> ("C" for "choice") and my "action" is simply a choice of measuring the box system in the state |Cab> or in the state |Cb> and not a mixture of them.

If Omega sets up an EPR-like entanglement between its variable and the box system of the form m|Oab>|Cab> + n|Ob>|Cb>, and then chooses to measure a mixed state of its variable, say, |Oab>+|Ob>, it can bifurcate the universe. Then, if I measure |Cab> (i.e. choose A+B), I end up in the same universe as the one in which Omega measured its variable to be |Oab> and if I choose |Cb>, I end up in the same universe as the one in which Omega measured its variable to be |Ob>. Therefore, if our two systems are entangled this way, Omega wouldn't need to take any trouble to simulate me at all in order to ensure its reputation of being a perfect predictor!

That is only as far as Omega's reputation for being a perfect predictor is concerned. But hold on for a moment there. In this setup, the box system's state is not disconnected from that of Omega's predictor variable even if Omega has left the galaxy and yet Omega cannot causally influence it "contents". In my thinking, this is an argument against the stance of the "causal decision theorists" that whatever the contents of the box, it is "fixed" and therefore I maximize my utility by picking A+B. This is now an argument for the one boxers observing that Omega has shown a solid history of being right (i.e. Omega's internal variable has always correlated with the choices of all the people before), forming the simplest (?) explanation that Omega could be using quantum entanglement (edit: EPR-like entanglement) to effect the correlation, and therefore choosing to one box so that they end up in the universe with a million bucks instead of the one with a thousand.

So, my final question to people here is this - does knowledge of quantum physics resolve Newcomb's problem in favour of the one boxers? If not, the arguments certainly would be interesting to read :)

edit: To clarify the argument against the causal decision theorists, "B is either empty or has a million bucks" is not true. It could be in a superposition of the two that is entangled with Omega's variable. Therefore the standard causal argument for picking A+B doesn't hold any 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 here is build it and let it solve all problems like these. That way, you win, at least in the time-management sense. Well, you might lose if you encountered Omega before the FAI was up and running, but that seems unlikely. Am I missing something here?

It will also have to precommit to mere humans who can't read its source code and can't predict the future, so solving the problem in the case where you meet Omega doesn't solve the problem in general.

Newcomb's Problem is silly. It's only controversial because it's dressed up in wooey vagueness. In the end it's just a simple probability question and I'm surprised it's even taken seriously here. To see why, keep your eyes on the bolded text:

Omega has been correct on each of 100 observed occasions so far - everyone [on each of 100 observed occasions] 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.

What can we anticipate from the bolded part? The only actionable belief we have at this point is that 100 out of 100 times, one-boxing made the one-boxer rich. The details that the boxes were placed by Omega and that Omega is a "superintelligence" add nothing. They merely confuse the matter by slipping in the vague connotation that Omega could be omniscient or something.

In fact, this Omega character is superfluous; the belief that the boxes were placed by Omega doesn't pay rent any differently than the belief that the boxes just appeared at random in 100 locations so far. If we are to anticipate anything different knowing it was Omega's doing, on what grounds? It could only be because we were distracted by vague notions about what Omega might be able to do or predict.

The following seemingly critical detail is just more misdirection and adds nothing either:

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.

I anticipate nothing differently whether this part is included or not, because nothing concrete is implied about Omega's predictive powers - only "superintelligence from another galaxy," which certainly sounds awe-inspiring but doesn't tell me anything really useful (how hard is predicting my actions, and how super is "super"?).

The only detail that pays any rent is the one above in bold. Eliezer is right that one-boxing wins, but all you need to figure that out is Bayes.

EDIT: Spelling

print "I will run some trial games (at least 5) to calibrate the predictor."
print ("As soon as the predictor will reach the expected quality level\n"
      "I will run the actual Newcomb game. Be warned you won't be\n" 
      "warned when calibration phase will end and actual game begin\n"
      "this is intended to avoid any perturbation of predictor accuracy.\n")

# run some prelude games (to avoid computing averages on too small a set)
# then compute averages to reach the intended prediction quality
# inecting some randomness in prelude and precision quality avoid
# anybody (including program writer) to be certain of when
# calibration ends. This is to avoid providing to user data that
# will change it's behavior and defeats prediction accuracy.
import random
# 5 to 25 calibration move
prelude = (5 + random.random() * 20.0)              
# 90% accuracy or better, and avoid infinite loop
# we do not tell how much better to avoid guessers
accuracy = 1.0 - (random.random() * 0.1) - 0.01 
# postlude is the number of test games where desired accuracy must be kept
# before running the actual game
# postlude will be a random number between 1 and 5 to avoid players guessing
# on the exact play time when percent will change, this could give them some
# hint on the exact final game time. It is possible the current postlude 
# can still be exploited to improve cheater chances above intended predictor
# values, but it's just here to get the idea... and besides outguessing omega
# the cheater is only doing so in the hope of getting 100 bucks.
# How much energy does that deserve ?
postlude = 0 
one = total = two = 0
while ((total < prelude) and (int(postlude) != 1)):
    a = raw_input ("1 - One-box, 2 - Two-boxes : ")
    if not a in ['1', '2']: continue
    if a == '1':
        one += 1
    else:
        two += 1
    total += 1
    print "current accuracy is %d%%" % int(100.0 * max(two, one) / total)
    if (max(two, one) * 1.0 < total * accuracy):
        if postlude != 0 :
            postlude -= 1
        else:
            postlude = 1 + random.random() * 5.0
    else:
        postlude = 0

# Now prediction accuracy is good enough, run actual Newcomb's game
# prediction is truly a prediction of the future
# nothing prevents the user to choose otherwise.
#print "This is the actual Newcomb game, but I won't say it"
prediction = 1 if one > two else 2
finished = False
while not finished:
    a = raw_input ("1 - One-box, 2 - Two-boxes : ")
    if a == '1':
        if prediction == 1:
            print "You win 1 000 000 dollars"
        else:
            print "You win zero dollars"
        finished = True
    elif a == '2':
        if prediction == 1:
            print "You win 1 000 100 dollars"
        else:
            print "You win 100 dollars"
        finished = True

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.

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-boxing, then Omega will have predicted that. It doesn't matter whether or not Omaga left already and box 1 already either contains 1M$ or 0$. No matter how long I ruminate back and forth, if I two-box I've lost because Omega is a perfect predictor and would thus have predicted it.

If Omega indeed has all the properties that are claimed, then there are only two possible outcomes: If you take one box, you'll get 1M$, if you take two, then you get 1000$. It is true, that box 1 either contains 1M$ or nothing by the time Omega left - but what the box contains is still 100% correlated with my upcoming final decision and nothing is going to change that. End of story. Ergo, CDT is wrong and a model that's at odds with reality.

PS: Interestingly, if opening the lid on these boxes is the trigger moment that counts as a "decision", you could just put the opaque box into an X-ray and this act alone would instantly transform Omega into a liar, regardless of whether it contained 1M$ or nothing. It couldn't possibly show an empty box without making Omega a liar, because contrary to what it said I could no longer actually decide to open only box 1 and get the 1M$. Conversely, if the box does contain 1M$, then I could just two-box, making Omega a liar with respect to its prediction.

So Omega would HAVE TO specifically forbid peeping into the opaque box. If it didn't do that, Omega would risk being a liar one way or another, once I looked into the 1st box without opening it and either found 1M$ or nothing.

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 encounter situations in which your decision can affect the prior probabilities, but it's no mystery what to do when that situation arises--the same thing as always, maximize that utility function.

Of course, while I can be proud of myself for spotting that right away, I can't be too proud because I know I was helped a lot by the fact that my mind was in a "thinking about Eliezer Yudkowsky" mode already, a mode it's not necessarily in by default and might not be when I am presented with a dilemma (unless I make a conscious effort to put it there, which I guess now I stand a better chance of doing). I was expecting for a Bayesian solution to the problem and spotted it even though it wasn't even the point of the example. I've seen this problem before, after all, without the context of being brought up by you, and I certainly didn't come up with that solution at the time.

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:
  1000000 P(C) = 1000 + 1000000 (1 - P(C))
  1000 P(C) = 1001 - 1000 P(C)
  2000 P(C) = 1001
  P(C) = 1001/2000 = 0.5005 = 50.05%

So as long as you are at least 50.05% sure that Omega's model of the universe describes you accurately, you should pick the one box. It's a little confusing because it seems like cause precedes effect in this situation, but that's not the case; your behaviour affects the behaviour of a simulation of you. Assuming Omega is always right: if you take one box, then you are the type of person who would take the one box, and Omega will see that you are, and you will win. So it's the clear choice.

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 themselves. If you know they are predicting you, and will act in a certain way accordingly, the rational solution is simply to follow through on whichever prediction is most profitable, as if they could actually see the future to make such a prediction correctly. Since you're deterministic, that you will do this is predictable, and thus, the prediction is self-fulfilling.

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 rational for two-boxing. Be it a type-1 or type-2 reason, in both cases, Omega is able to reward me for one-boxing if that what he wants, and with 100 prior cases, he really seems to be wanting that.

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 idealized and contradictory one - basically the contradiction between the reports of 100 accurate trials by Omega, the assumption that there was no cheating involved, the assumption about no reverse time causality, etc. If you do so, your priors will tell you the right answer.

Ha - although I expect your belief in forward time causality is higher than your confidence in your use of Jaynes formalism.

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 you can precommit yourself before Omega examines you; then you are directly causing box B to be filled."

Is this your objection? The problem is, you don't know if the superintelligent alien is basing anything on "precommital." Maybe the superintelligent alien has some technology or understanding that allows him to actually see the end result of your future contemplation. Maybe he's solved time travel and has seen what you pick.

Unless you understand not only the alien's mode of operation but also his method, you really are just guessing at how he'll decide what to put in box two. And your record on guesses is not as good as his.

There's nothing mystical about it. You do it because it works. Not because you know how it works.

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 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.

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 to me. A one boxer should say, if consistent, sure, 100 people saw it it is true. No matter what priors we put on the resurrection of Jesus being true.

To me, it is incredibly more likely that either people are lying to me, or at least being wrong. I have seen magicians make things appear and disappear in boxes that were already sealed, after they left. It is WAY more likely that this is some kind of test and/or scam.

Which is not to say I wouldn't one-box, I would! Whatever scam Omega is running, I'd rather have the million dollars, or prove Omega a fraud by finding an empty box, then to have only $1000, or prove Omega wrong by finding a full box and having $1001000.

And this is precisely what I would announce to the people before publicly opening the one box, and this is, if it is not a fraud, Omega would have known I would do.

As ot 100 times to prove something that unlikely? Siegrfied and Roy have made thousands of tigers appear and disappear in cages they could not have had sufficient access too. As odd as they are, it is unlikely (IMHO) that they are superintelligent aliens.

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 to condition yourself in advance so that you will take 1 box in Newcomb-type situations, and ideally do so quite reflexively, without even thinking about it. That way, Omega will predict (correctly) that you will 1-box, and you'll get the $1 million.

This is fairly close to the standard analysis, though what I'd dispute about the standard version is that there is anything "irrational" in so-conditioning oneself. It seems to me that we train ourselves all the time to do things without thinking about them (such as walking, driving to work, typing out letters to spell words etc) and it's perfectly reasonable for us to do that where it will have higher expected utility for us.

There might even be a significant practical issue here: quite possibly a lot of moral discipline involves conditioning of oneself in advance to do things which don't (at that time) maximise utility. This is so we actually get to be put in positions of responsibility, where being in such positions has higher utility than not being in them - real-life Newcomb problems.In practice, we seem to be quite good at approximating Omega with each other on a social level; when hiring a security guard for instance, we seem to be quite good at predicting who will defend our property rather than run off with it. Not perfect of course.

 Box B filled.               Box B empty.

      0.                          0.99.               Alf 2 boxes

      0.                          0.01.               Alf 1 boxes

   Box B filled.          Box B empty.

   0.099.                     0.891.                    Alf 2 boxes

   0.009.                     0.001.                    Alf 1 boxes

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 about your original argument that is different in the transparent box case?

If you're really a one boxer, that means you can look at an empty box and still pick it.

I was surprised that the rec.puzzles FAQ answer to this doesn't appear in the replies. (Maybe it's here and I just missed it.)

While you are given that P(do X | predict X) is high, it is not given that P(predict X | do X) is high. Indeed, specifying that P(predict X | do X) is high would be equivalent to specifying that the being could use magic (or reverse causality) to fill the boxes. Therefore, the expected gain from either action cannot be determined from the information given.

In other words, we can't tell if (how much) our actions determine the outcome, so we can't make a rational decision.

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 know the mechanism with sufficient certainty that I should reason myself into two-boxing against the evidence to date.

Does it matter which undetectable unbelievable process Omega is using for me to pick my strategy? I don't think it does - I have to acknowledge that I'm out of my depth with this alien and arguments against causality defiance or the impossibility of undetectable money vaporisers are not going to help me take the million.

Another tack: Omega isn't a super intelligence - he's got a ship, a plan, and a lot of time on his hands. He turns up on millions of worlds to play this game. His guesses are pretty lousy, he guesses right only x percent of the time. We are the only planet on which he's consistently guessed right. We don't know what x is in the full sample size. Looking at what his results are here, it looks good. Does it really seem rational to second guess the sample we see?

It seems to me that we have to accept some pretty wild statements and then start reasoning based on them for us to come to a losing strategy. If we doubt the premises to some degree then does it become clear that the most reasonable strategy is one-boxing?

This reminds me eerily of the Calvinist doctrine of predestination. The money is already there, and making fun of me for two-boxing ain't gonna change anything.

A question - how could Omega be a perfect predictor, if I in fact have a third option - namely leaving without taking either box? This possibility would, in any real-life situation, lead me to two-box. I know this and accept it.

Then there's always the economic argument: If $1000 is a sum of money that matters a great deal to me, I'm two-boxing. Otherwise, I'd prefer to one-box.

So, I'm sure this isn't an original thought but there are a lot of comments and my utility function is rolling its eyes at the thought of going through them all to see whether this comment is redundant, as compared to writing the comment given I want to sort my thoughts out verbally anyway.

I think the standard form of the question should be changed to the one with the asteroid. Total destruction is total destruction, but money is only worth a) what you can buy with it and b) the effort it takes to earn it.

I can earn $1000 in a month. Some people could earn it in a week. What is the difference between $1m and $1m + $1000? Yes, it's technically a higher number, but in terms of my life that is not a statistically significant difference. Of course I'd rather definitely have $1m than risk having nothing for the possibility of having $1m + $1000.

The causal decision theory versions of this problem don't look ridiculous because they take the safe option, they look ridiculous because the utility of two-boxing is not significant in comparison with the potential utility of one-boxing. That is, a one-boxer doesn't lose much if they're wrong: IF box 2 already contained nothing when they chose it, they only missed their chance at $1000, whereas IF box 2 already contained $1m a two-boxer misses their chance at $1m.

Obviously an advanced decision theory needs a way to rank the potential risks - if you postulate it as the asteroid, the risk is much more concrete.

I'd just like to note that as with most of the rationality material in Eliezer's sequences, the position in this post is a pretty common mainstream position among cognitive scientists. E.g. here is Jonathan Baron on page 61 of his popular textbook Thinking and Deciding:

the best kind of thinking, which we shall call rational thinking, is whatever kind of thinking best helps people achieve their goals. If it should turn out that following the rules of formal logic leads to eternal happiness, then it is rational thinking to follow the laws of logic (assuming that we all want eternal happiness). If it should turn out, on the other hand, that carefully violating the laws of logic at every turn leads to eternal happiness, then it is these violations that we shall call rational.

This view is quoted and endorsed in, for example, Stanovich 2010, p. 3.

It seems to me that if you make a basic bayes net with utilities at the end. The choice with the higher expected utility is to one box. Say:
P(1,000,000 in box b and 10,000 in box a|I one box) = 99%
P(box b is empty and 10,000 in box a|I two box) = 99%
hence
P(box b is empty and 10,000 in box a|I one box) = 1%
P(1,000,000 in box b and 10,000 in box a|I two box) = 1%
So
If I one box i should expect 99%1,000,000+1%0 = 990,000
If I two box i should expect 99%10,000+1%1,010,000 = 20,000
Expected utility(I one box)/Expected utility(I two box) = 49.5, so I should one box by a land slide. This is assuming that omega has a 99% rate of true positive, and of true negative; it's more dramatic if we assume that omega is perfect. If P(1,000,000 in box b and 10,000 in box a|I one box) = P(box b is empty and 10,000 in box a|I two box) = 100%, then Expected utility(I one box)/Expected utility(I two box) = 100. If omega is perfect, by my calculation we should expect one boxing to be a 100 times more profitable than two boxing.

This is the sort of math I usually use to decide. Is this none-standard, did I make a mistake, or does this method produce stupid results elsewhere?

I think you went wrong when you said:

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.

because Omega doesn't reward people for their choice to pick box B, he rewards them for being implementations of any of the many algorithms that would pick box B.

I think that the causal decision theory algorithm is the winning way for problems where your mind is not read (when you take into account that causal decision theory can be swayed to make choices so as deceive others about your real algorithm). Problems where your mind is read do not usually show up in real life. I think there is no winning way for conceivable universes in general, so I want to be an implementation of the winning algorithm for this universe, which seems to be causal decision theory.

Sorry, I'm new here, I am having trouble with the Idea that anyone would consider taking both boxes in a real world situation. How would this puzzle be modeled differently, versus how would it look differently if it were Penn and Teller flying Omega?

If Penn and Teller were flying Omega then they would have been able to produce exactly the same results as seen, without violating causality or time travelling or perfectly predicting people by just cheating and emptying the box after you choose to take both.

Given that "it's cheating" is a significantly more rational idea than "it's smart enough to predict 100 people" in terms of simplicity and results seen, why not go with that as a rational reason to pick just box B? The only reason one would take both is if it proved it was not cheating, how it could do that without also convincing me of its predictive powers I don't know, and once convinced of is predictive powers I would have to take Box B.

So taking both boxes only makes sense if you know it is not cheating, and know it can be wrong. I notice I am confused, how can you both know it is not cheating, and not know that it is correct in it's prediction.

I think that the reason this puzzle begets irrationality is that one of the fundamental things you must do to parse the puzzle is irrational, that is 'believe that the machine is not cheating', given the alternatives and no further info.