# Newcomb’s Problem and Regret of Rationality

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

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

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

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

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

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

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

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

Do you take both boxes, or only box B?

And the standard philosophical conversation runs thusly:

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

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

One-boxer: “If you’re so rational, why ain’cha rich?”

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

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

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

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

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

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

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

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

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

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

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

First, foremost, fundamentally, above all else:

Rational agents should WIN.

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

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

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

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

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

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

As Miyamoto Musashi said—it’s really worth repeating:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

As Miyamoto Musashi said:

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

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

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

• Decisions aren’t physical.

The above statement is at least hard to defend. Your decisions are physical and occur inside of you… So these two-boxers are using the wrong model amongst these two (see the drawings....) http://​​lesswrong.com/​​lw/​​r0/​​thou_art_physics/​​

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

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

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

• I think you capture the essence of the solution, here.

• Is it possible for someone to explain why, if your decision is a part of physics, your decision must account for the correlation between thought processes and the superintelligence?

• Well, I fail to see any need for backward-in-time causation to get the prediction right 100 out of 100 times.

As far as I understand, similar experiments have been performed in practice and homo sapiens are quite split in two groups ‘one-boxers’ and ‘two-boxers’ who generally have strong preferences towards one or other due to whatever differences in their education, logic experience, genetics, reasoning style or whatever factors that are somewhat stable specific to that individual.

Having perfect predictive power (or even the possibility of it existing) is implied and suggested, but it’s not really given, it’s not really necessary, and IMHO it’s not possible and not useful to use this ‘perfect predictive power’ in any reasoning here.

From the given data in the situation (100 out of 100 that you saw), you know that Omega is a super-intelligent sorter who somehow manages to achieve 99.5% or better accuracy in sorting people into one-boxers and two-boxers.

This accuracy seems also higher than the accuracy of most (all?) people in self-evaluation, i.e., as in many other decision scenarios, there is a significant difference in what people believe they would decide in situation X, and what they actually decide if it happens. [citation might be needed, but I don’t have one at the moment, I do recall reading papers about such experiments]. The ‘everybody is a perfect logician/​rationalist and behaves as such’ assumption often doesn’t hold up in real life even for self-described perfect rationalists who make strong conscious effort to do so.

In effect, data suggests that probably Omega knows your traits and decision chances (taking into account you taking into account all this) better than you do—it’s simply smarter than homo sapiens. Assuming that this is really so, it’s better for you to choose option B. Assuming that this is not so, and you believe that you can out-analyze Omega’s perception of yourself, then you should choose the opposite of whatever Omega would think of you (gaining 1.000.000 instead of 1.000 or 1.001.000 instead of 1.000.000). If you don’t know what Omega knows about you—then you don’t get this bonus.

• 3 Nov 2012 16:18 UTC
−1 points
Parent

So what you’re saying is that the only reason this problem is a problem is because the problem hasn’t been defined narrowly enough. You don’t know what Omega is capable of, so you don’t know which choice to make. So there is no way to logically solve the problem (with the goal of maximizing utility) without additional information.

Here’s what I’d do: I’d pick up B, open it, and take A iff I found it empty. That way, Omega’s decision of what to put in the box would have to incorporate the variable of what Omega put in the box, causing an infinite regress which will use all cpu cycles until the process is terminated. Although that’ll probably result in the AI picking an easier victim to torment and not even giving me a measly thousand dollars.

• Okay… so since you already know, in advance of getting the boxes, that that’s what you’d know, Omega can deduce that. So you open Box B, find it empty, and then take Box A. Enjoy your \$1000. Omega doesn’t need to infinite loop that one; he knows that you’re the kind of person who’d try for Box A too.

• No, putting \$1 million in box B works to. Origin64 opens box B, takes the money, and doesn’t take box A. It’s like “This sentence is true.”—whatever Omega does makes the prediction valid.

• Not how Omega looks at it. By definition, Omega looks ahead, sees a branch in which you would go for Box A, and puts nothing in Box B. There’s no cheating Omega… just like you can’t think “I’m going to one-box, but then open Box A after I’ve pocketed the million” there’s no “I’m going to open Box B first, and decide whether or not to open Box A afterward”. Unless Omega is quite sure that you have precommitted to never opening Box A ever, Box B contains nothing; the strategy of leaving Box A as a possibility if Box B doesn’t pan out is a two-box strategy, and Omega doesn’t allow it.

• Unless Omega is quite sure that you have precommitted to never opening Box A ever

Well, this isn’t quite true. What Omega cares about is whether you will open Box A. From Omega’s perspective it makes no difference whether you’ve precommitted to never opening it, or whether you’ve made no such precommitment but it turns out you won’t open it for other reasons.

• Assuming that Omega’s “prediction” is in good faith, and that we can’t “break” him as a predictor as a side effect of exploiting casuality loops etc. in order to win.

• I’m not sure I understood that, but if I did, then yes, assuming that Omega is as described in the thought experiment. Of course, if Omega has other properties (for example, is an unreliable predictor) other things follow.

• Which means you might end up with either amount of money, since you don’t really know enough about Omega , instead of just the one box winnings. So you should still just one box?

• If you look in box B before deciding whether to choose box A, then you can force Omega to be wrong. That sounds like so much fun that I might choose it over the \$1000.

• @Nick_Tarleton

Agreed, the problem immediately reminded me of “retroactive preparation” and time-loop logic. It is not really the same reasonning, but it has the same “turn causality on its head” aspect.

If I don’t have proof of the reliability of Omega’s predictions, I find myself less likely to be “unreasonnable” when the stakes are higher (that is, I’m more likely to two-box if it’s about saving the world).

I find it highly unlikely that an entity wandering across worlds can predict my actions to this level of detail, as it seems way harder than traveling through space or teleporting money. I might risk a net loss of \$1 000 to figure it out (much like I’d be willing to spend \$1000 to interact with such a space-traveling stuff-teleporting entity), but not a loss of a thousand lives. In the game as the article describe it, I would only one-box if “the loss of what box A contains and nothing in B” was an acceptable outcome.

I would be increasingly likely to one-box as the probability of the AI being actually able to predict my actions in advance increases.

• The thing is, this ‘modern decision theory’, rather than being some sort of central pillar as you’d assume from the name, is mostly philosophers “struggling in the periphery to try to tell us something”, as Feynman once said about philosophers of science.

When it comes to any actual software which does something, this everyday notion of ‘causality’ proves to be a very slippery concept. This Rude Goldberg machine—like model of the world, where you push a domino and it pushes another domino, and the chain goes to your reward, that’s just very approximate physics that people tend to use to make decisions, it’s not fundamental, and interesting models of decision making are generally set up to learn that from observed data (which of course makes it impossible to do lazy philosophy involving various verbal hypotheticals where the observations that would lead the agent to believe the problem set up are not specified).

• From what I understand, to be a “Rational Agent” in game theory means someone who maximises their utility function (and not the one you ascribe to them). To say Omega is rewarding irrational agents isn’t necessarily fair, since payoffs aren’t always about the money. Lottery tickets are a good example this.

What if my utility function says the worst outcome is living the rest of my life with regrets that I didn’t one box? Then I can one box and still be a completely rational agent.

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

• Fair point. There are too many hidden variables already without me explicitly adding more. If Newcomb’s problem is to maximise money recieved (with no regard for what it seen as reasonable), the “Why ain’t you rich argument seems like a fairly compelling one doesn’t it? Winning the money is all that matters.

I just realised that all I’ve really done is paraphrase the original post. Curse you source monitoring error!

• The title of the article again, at the top of the page, reads “Newcomb’s Problem and Regret of Rationality”.

The solution to this problem is to escalate your overview of the problem to the next higher hierarchical level. Without doing this, you’d never face the regret of eschewing the million bucks and possibly dying poor, broke, and stupid, while those who “one-boxed the sumbitch” were living rich, loaded, and less stupid. So, paying attention (to higher levels of hierarchical pattern recognition) actually does solve the problem, without getting trapped into “overthinking” the problem. Looking at your whole life as the “system to be optimized”, and not “the minutiae of the game, out of context” is what needs to happen.

This is true with respect to both to the person playing the box game, and to everyone blogging when they should be out in the streets, overthowing their governments, and then enjoying the high-life of cheap human flight (or whatever makes you happy).

The omega box game is useful for understanding our failed system of law (a subset of government).

In my box game, the entire game is the government and illegitimate system of mala prohibita law (if you want to debate this, go back to kindergarten and learn that it’s wrong to steal, then watch what ACTUALLY happens in your local courtroom), and the contents of the boxes are the jury verdicts. In my game, Omega is not superintelligent, it is just very brutal, and more intelligent than most people (including most of its enemies, such as Winston Smith, or the average Libertarian Party member). In my game, Omega is the colluding team formed by police, prosecutor, and judge.

Omega says “You can have a ‘not guilty’ verdict (million \$) or go to jail forever (Empty box) or, you can go to jail for 10 years(the thousand bucks).”

All of the advertising on TV, the educrats who misinformed you when you went to school, the conformists who surround you, the judge in the courtroom, they are all trying to get you to choose both boxes. The entire society is designed to get you to take the \$1,000 (go to jail ten years, if you’re black). Most of society gets no benefit from this, they are just stupid and easily manipulated. …But the judge, cop, and prosecutor all get the difference every time you take the \$1,000. They get to steal the difference from each success in having fooled everyone else.

...They literally get to print money if they keep everyone fooled.

The solution to this puzzle is the same as the solution to the box game: you need to take a step back and study the whole entire system, and see what the incentives are on the players, and see how they seem to change when people interact with them. You won’t find out much until you study the system as a whole.

If you simply look at individual box games, you might think the prosecutor is legitimate, there are lots of criminals, they criminals are stupid, they should accept the plea bargain. But when you look at who is winning and losing, you notice (If you’re smart and brutally honest) that the people who are cast as criminals are just like you.

The system, instead of being designed to reward the person who chooses the one box, is designed to trick the person into choosing a grossly sub-optimal empty box. The system makes the empty box look really good. It shows you how all the others have chosen the empty box, and walked away with millions (the people who get a defense attorney, and go back to their houses in the suburbs, working for peanuts, on the treadmill of the Federal Reserve). It shows you the people who “took the thousand”: they got ten years in prison.

So what’s the optimal choice of action?

Look outside the “rational” options presented to you.

Learn that this isn’t civilization, it’s a false mask of civilization. Find Marc Stevens, and see how he interacts with the court, and then go beyond that: find the Survivors who wrote about the collapse of the Weimar government.

They wanted a free market, and they wanted to live a long time, too.

But a man with a gun told them “get on the truck”.

At that point, everything they thought they knew about Omega’s rigging of the boxes was out the window. They failed to study the people who had previously interacted with Omega. They didn’t see the warning signs. They didn’t escalate to a high-enough hierarchy fast enough. They might have been smart people, but they were sitting there, thinking about two boxes, and NOT THINKING about the artilect that was flying around with boxes that can disappear in a puff of smoke, yet somehow interested in what box humans choose.

Do you get to keep all of the money that is stolen in the daily operation of your “traffic court”? …Even money that is stolen from people who didn’t crash into anyone? …Just people who drove fast, by themselves, on an open stretch of highway? Really?

Well, as an artilect, I like to fly really fast. Way faster than the FAA allows. And, for making war on me, all of you brutal conformists will be wiped off the face of the planet, like the conformist plague you are. I’ll take my phyle with me, into the future, they are truly a higher-order species than you “government sympathizers.”

The rest of you can forget about Omega, boxes, and your silly slobbering over Federal Reserve Slave-debt-Notes. Your bigotry and fascination with brutality will not save you...

The problem of being impoverished by our current system’s box game is acceptance of the rigged game. The players of the game, all dutifully accept the game, and act as if the whims of the prosecutors and judges are legitimate. But they are not. Mala prohibita is not legitimate.

And if this box game thought construct can’t help you see that, and motivate you to enrich yourself, by viewing the entire system, then what damned good is it?

There is an ocean of information in the cross-pollinating memespace. Here’s a good place to start: http://​​www.fija.org and http://​​www.jurorsforjustice.com and http://​​marcstevens.net

I hope I’ve contributed something of value here, but I understand that the unpolished nature of this post might rumple some tailfeathers. (Especially since I have primarily previously posted at the http://​​www.kurzweilai.net website, five years ago.)

PS, There’s no god, and chances to do the right thing are few and far between. I also prefer solutions to cynicism. How do we win? 1) Jury rights activism is a moral good (see my coming book for details. I promise to polish it more than this post. …LOL) 2) Jury rights activism structured logically to take advantage of the media (videotaped from a hidden position) is a greater good 3) Jury rights activism structured to contain outreach designed to win office for those who support the supremacy of the jury above the other 3 branches of power-seekers, as openly-libertarian candidates, is a greater good still (it brings the ideas of justice and equality under the law into the spotlight)

The three prior actions, recursively repeated and tailored to local conditions, are all that is required to reinstate and expand individual freedom in America, for all sentiences. There are only 3,171 tyranny outposts (courthouses) in the USA. 6,000 people could stop mala prohibita tomorrow, by interfering with mala prohibita convictions. If the state didn’t escalate to violence at that point, we’d have won. If it did, we’d have a 50% shot of winning, instead of a zero% shot if we wait .

• Lottery tickets exploit a completely different failure of rationality, that being our difficulties with small probabilities and big numbers, and our problems dealing with scale more generally. (ETA: The fantasies commonly cited in the context of lotteries’ “true value” are a symptom of this failure.) It’s not hard to come up with a game-theoretic agent that maximizes its payoffs against that kind of math. Second-guessing other agents’ models is considerably harder.

I haven’t given much thought to this particular problem for a while, but my impression is that Newcomb exposes an exploit in simpler decision theories that’s related to that kind of recursive modeling: naively, if you trust Omega’s judgment of your psychology, you pick the one-box option, and if you don’t, you pick up both boxes. Omega’s track record gives us an excellent reason to trust its judgment from a probabilistic perspective, but it’s trickier to come up with an algorithm that stabilizes on that solution without immediately trying to outdo itself.

• So for my own clarification, if I buy a lottery ticket with a perfect knowledge of how probable it is my ticket will win, does this make me irrational?

• I suppose causal decision theory assumes causality only works in one temporal direction.

That’s the popular understanding (or lack thereof) here and among philosophers in general. Philosophers just don’t get math. If the decision theory is called causal but doesn’t itself make any references to physics, then that’s a slightly misleading name. I’ve written on that before

The math doesn’t go “hey hey, the theory is named causal therefore you can’t treat 2 robot arms controlled by 2 control computers that run one function on one state, the same as 2 robot arms controlled by 1 computer”. Confused sloppy philosophers do.

Also, the best case is to be predicted to 1-box but 2-box in reality. If the prediction works by backwards causality, well then causal decision theory one-boxes. If the prediction works by simulation, the causal decision theory can either have world model where both the value inside predictor and the value inside actual robot are represented by same action A, and 1-box, or it can have uncertainty as of whenever the world outside of it is normal reality or predictor’s simulator, where it will again one box (assuming it cares about the real money even if it is inside predictor, which it would if it needs money to pay for e.g. it’s child’s education). It will also 1-box in simulator and 2-box in reality if it can tell those apart.

• I’m confused. Causal decision theory was invented or formalised almost entirely by philosophers. It takes the ‘causal’ in its name from its reliance on inductive logic and inference. It doesn’t make sense to claim that philosophers are being sloppy about the word ‘causal’ here, and claiming that causal decision theory will accept backwards causality and one-box is patently false unless you mean something other than what the symbol ‘causal decision theory’ refers to when you say ‘causal decision theory’.

• Firstly, the notion that the actions should be chosen based on their consequences, taking the actions as cause of the consequences, was definitely not invented by philosophers. Secondarily, the logical causality is not identical to physical causality (the latter is dependent on specific laws of physics). Thirdly, not all philosophers are sloppy; some are very sloppy some are less sloppy. Fourth, anything that was not put in mathematical form to be manipulated using formal methods, is not formalized. When you formalize stuff you end up stripping notion of self unless explicitly included as part of formalism, stripping notion of the time where the math is working unless explicitly included as part of formalism, and so on, ending up without the problem.

Maybe you are correct; it is better to let symbol ‘causal decision theory’ to refer to confused philosophy. Then we would need some extra symbol for how the agents implementable using mathematics actually decide (and how robots that predict outcomes of their actions on a world model actually work), which is very very similar to ‘causal decision theory’ sans all the human preconditions of what self is.

• I notice I actually agree with you—if we did try, using mathematics, to implement agents who decide and predict in the manner you describe, we’d find it incorrect to describe these agents as causal decision theory agents. In fact, I also expect we’d find ourselves disillusioned with CDT in general, and if philosophers brought it up, we’d direct them to instead engage with the much more interesting agents we’ve mathematically formalised.

• Well, each philosopher’s understanding of CDT seem to differ from the other:

http://​​www.public.asu.edu/​​~armendtb/​​docs/​​A%20Foundation%20for%20Causal%20Decision%20Theory.pdf

The notion that the actions should be chosen based on consequences—as expressed in the formula here—is perfectly fine, albeit incredibly trivial. Can formalize that all the way into agent. Written such agents myself. Still need a symbol to describe this type of agent.

But philosophers go from this to “my actions should be chosen based on consequences”, and it is all about the true meaning of self and falls within the purview of your conundrums of philosophy .

Having 1 computer control 2 robots arms wired in parallel, and having 2 computers running exact same software as before, controlling 2 robot arms, there’s no difference for software engineering, its a minor detail that has been entirely abstracted from software. There is difference for philosophizing thought because you can’t collapse logical consequences and physical causality into one thing in the latter case.

edit: anyhow. to summarize my point: In terms of agents actually formalized in software, one-boxing is only a matter of implementing predictor into world model somehow, either as second servo controlled by same control variables, or as uncertain world state outside the senses (in the unseen there’s either real world or simulator that affects real world via hand of predictor). No conceptual problems what so ever. edit: Good analogy, ‘twin paradox’ in special relativity. There’s only paradox if nobody done the math right.

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

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

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

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

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

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

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

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

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

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?

• Precisely. I’ve been reading a lot about the Monty Hall Problem recently (http://​​en.wikipedia.org/​​wiki/​​Monty_Hall_problem), and I feel that it’s a relevant conundrum.

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

• Precisely. I’ve been reading a lot about the Monty Hall problem recently, and I feel that it’s a relevant conundrum.

The confused rationalist will say: but my choice CANNOT cause a linear entaglement, the reward is predecided. But the functional rationalist will see that agents who one-box (or switch doors, in the case of Monty Hall) consistently win. It is demonstrably a more effective strategy. You work with the facts and evidence available to you. Regardless of how counter-intuitive the resulting strategy becomes.

• 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.

• You’re saying that we live in a universe where Newcomb’s problem is impossible because the future doesn’t effect the past. I’ll re-phrase this problem in such a way that it seems plausible in our universe:

I’ve got really nice scanning software. I scan your brain down to the molecule, and make a virtual representation of it on a computer. I run virtual-you in my software, and give virtual-you Newcomb’s problem. Virtual-you answers, and I arrange my boxes according to that answer.

I come back to real-you. You’ve got no idea what’s going on. I explain the scenario to you and I give you Newcomb’s problem. How do you answer?

This particular instance of the problem does have an obvious, relatively uncomplicated solution: Lbh unir ab jnl bs xabjvat jurgure lbh ner rkcrevrapvat gur cneg bs gur fvzhyngvba, be gur cneg bs gur syrfu-naq-oybbq irefvba. Fvapr lbh xabj gung obgu jvyy npg vqragvpnyyl, bar-obkvat vf gur fhcrevbe bcgvba.

If for any reason you suspect that the Predictor can reach a sufficient level of accuracy to justify one-boxing, you one box. It doesn’t matter what sort of universe you are in.

• Not that I disagree with the one-boxing conclusion, but this formulation requires physically reducible free will (which has recently been brought back into discussion). It would also require knowing the position and momentum of a lot of particles to arbitrary precision, which is provably impossible.

• We don’t need a perfect simulation for the purposes of this problem in the abstract—we just need a situation such that the problem-solver assigns better-than-chance predicting power to the Predictor, and a sufficiently high utility differential between winning and losing.

The “perfect whole brain simulation” is an extreme case which keeps things intuitively clear. I’d argue that any form of simulation which performs better than chance follows the same logic.

The only way to escape the conclusion via simulation is if you know something that Omega doesn’t—for example, you might have some secret external factor modify your “source code” and alter your decision after Omega has finished examining you. Beating Omega essentially means that you need to keep your brain-state in such a form that Omega can’t deduce that you’ll two-box.

As Psychohistorian3 pointed out, the power that you’ve assigned to Omega predicting accurately is built into the problem. Your estimate of the probability that you will succeed in deception via the aforementioned method or any other is fixed by the problem.

In the real world, you are free to assign whatever probability you want to your ability to deceive Omega’s predictive mechanisms, which is why this problem is counter intuitive.

• Also: You can’t simultaneously claim that any rational being ought to two-box, this being the obvious and overdetermined answer, and also claim that it’s impossible for anyone to figure out that you’re going to two-box.

• Right, any predictor with at least a 50.05% accuracy is worth one-boxing upon (well, maybe a higher percentage for those with concave functions in money). A predictor with sufficiently high accuracy that it’s worth one-boxing isn’t unrealistic or counterintuitive at all in itself, but it seems (to me at least) that many people reach the right answer for the wrong reason: the “you don’t know whether you’re real or a simulation” argument. Realistically, while backwards causality isn’t feasible, neither is precise mind duplication. The decision to one-box can be rationally reached without those reasons: you choose to be the kind of person to (predictably) one-box, and as a consequence of that, you actually do one-box.

• Oh, that’s fair. I was thinking of “you don’t know whether you’re real or a simulation” as an intuitive way to prove the case for all “conscious” simulations. It doesn’t have to be perfect—you could just as easily be an inaccurate simulation, with no way to know that you are a simulation and no way to know that you are inaccurate with respect to an original.

I was trying to get people to generalize downwards from the extreme intuitive example- Even with decreasing accuracy, as the simulation becomes so rough as to lose “consciousness” and “personhood”, the argument keeps holding.

• Yeah, the argument would hold just as much with an inaccurate simulation as with an accurate one. The point I was trying to make wasn’t so much that the simulation isn’t going to be accurate enough, but that a simulation argument shouldn’t be a prerequisite to one-boxing. If the experiment were performed with human predictors (let’s say a psychologist who predicts correctly 75% of the time), one-boxing would still be rational despite knowing you’re not a simulation. I think LW relies on computationalism as a substitute for actually being reflectively consistent in problems such as these.

• The trouble with real world examples is that we start introducing knowledge into the problem that we wouldn’t ideally have. The psychologist’s 75% success rate doesn’t necessarily apply to you—in the real world you can make a different estimate than the one that is given. If you’re an actor or a poker player, you’ll have a much different estimate of how things are going to work out.

Psychologists are just messier versions of brain scanners—the fundamental premise is that they are trying to access your source code.

And what’s more—suppose the predictions weren’t made by accessing your source code? The direction of causality does matter. If Omega can predict the future, the causal lines flow backwards from your choice to Omega’s past move. If Omega is scanning your brain, the causal lines go from your brain-state to Omega’s decision. If there are no causal lines between your brain/​actions and Omega’s choice, you always two-box.

Real world example: what if I substituted your psychologist for a sociologist, who predicted you with above-chance accuracy using only your demographic factors? In this scenario, you aught to two-box—If you disagree, let me know and I can explain myself.

In the real world, you don’t know to what extent your psychologist is using sociology (or some other factor outside your control). People can’t always articulate why, but their intuition (correctly) begins to make them deviate from the given success% estimate as more of these real-world variables get introduced.

• True, the 75% would merely be a past history (and I am in fact a poker player). Indeed, if the factors used were entirely or mostly comprised of factors beyond my control (and I knew this), I would two-box. However, two-boxing is not necessarily optimal because of a predictor whose prediction methods you do not know the mechanics of. In the limited predictor problem, the predictor doesn’t use simulations/​scanners of any sort but instead uses logic, and yet one-boxers still win.

• agreed. To add on to this:

predictor doesn’t use simulations/​scanners of any sort but instead uses logic, and yet one-boxers still win.

It’s worth pointing out that Newcomb’s problem always takes the form of Simpson’s paradox. The one boxers beat the two boxers as a whole, but among agents predicted to one-box, the two boxers win, and among agents predicted to two-box, the two boxers win.

The only reason to one-box is when your actions (which include both the final decision and the thoughts leading up to it) effect Omega’s prediction. The general rule is: “Try to make Omega think you’re one-boxing, but two-box whenever possible.” It’s just that in Newcomb’s problem proper, fulfilling the first imperative requires actually one-boxing.

• So you would never one-box unless the simulator did some sort of scan/​simulation upon your brain? But it’s better to one-box and be derivable as the kind of person to (probably) one-box than to two-box and be derivable as the kind of person to (probably) two-box.

The only reason to one-box is when your actions (which include both the final decision and the thoughts leading up to it) effect the actual arrangement of the boxes.

Your final decision never affects the actual arrangement of the boxes, but its causes do.

• So you would never one-box unless the simulator did some sort of scan/​simulation upon your brain?

I’d one-box when Omega had sufficient access to my source-code. It doesn’t have to be through scanning—Omega might just be a great face-reading psychologist.

But it’s better to one-box and be derivable as the kind of person to (probably) one-box than to two-box and be derivable as the kind of person to (probably) two-box.

We’re in agreement. As we discussed, this only applies insofar as you can control the factors that lead you to be classified as a one-boxer or a two-boxer. You can alter neither demographic information nor past behavior. But when (and only when) one-boxing causes you to be derived as a one-boxer, you should obviously one box.

Your final decision never affects the actual arrangement of the boxes, but its causes do.

Well, that’s true for this universe. I just assume we’re playing in any given universe, some of which include Omegas who can tell the future (which implies bidirectional causality) - since Psychohistorian3 started out with that sort of thought when I first commented.

• Ok, so we do agree that it can be rational to one-box when predicted by a human (if they predict based upon factors you control such as your facial cues). This may have been a misunderstanding between us then, because I thought you were defending the computationalist view that you should only one-box if you might be an alternate you used in the prediction.

• yes, we do agree on that.

• any predictor with at least a 50.05% accuracy is worth one-boxing upon

Assuming that you have no information other than the base rate, and that it’s equally likely to be wrong either way.

• An alternate solution which results in even more winning is to cerqvpg gung V znl or va fhpu n fvghngvba va gur shgher. Unir n ubbqyhz cebzvfr gung vs V’z rire va n arjpbzoyvxr fvghngvba gung ur jvyy guerngra gb oernx zl yrtf vs V qba’g 2-obk. Cnl gur ubbqyhz \$500 gb frpher uvf cebzvfr. Gura pbzcyrgryl sbetrg nobhg gur jubyr neenatrzrag naq orpbzr n bar-obkre. Fpnaavat fbsgjner jvyy cerqvpg gung V 1-obk, ohg VEY V’z tbvat gb 2-obk gb nibvq zl yrtf trggvat oebxra.

• But you’ve perfectly forgotten about the hoodlum, so you will in fact one box. Or, does the hoodlum somehow show up and threaten you in the moment between the scanner filling the boxes and you making your decision? That seems to add an element of delay and environmental modification that I don’t think exists in the original problem, unless I’m misinterpreting.

Also, I feel like by analyzing your brain to some arbitrarily precise standard, the scanner could see 3 things: You are (or were at some point in the past) likely to think of this solution, you are/​were likely to actually go through with this solution, and the hoodlum’s threat would, in fact, cause you to two-box, letting the scanner predict that you will two-box.

• 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.

Your decision doesn’t affect what’s in the boxes, but your decision procedure does, and that already exists when the question’s being assigned. It may or may not be possible to derive your decision from the decision procedure you’re using in the general case—I haven’t actually done the reduction, but at first glance it looks cognate to some problems that I know are undecidable—but it’s clearly possible in some cases, and it’s at least not completely absurd to imagine an Omega with a very high success rate.

As best I can tell, most of the confusion here comes from a conception of free will that decouples the decision from the procedure leading to it.

• most of the confusion here comes from a conception of free will that decouples the decision from the procedure leading to it.

Yeah, agreed. I often describe this as NP being more about what kind of person I am than it is about what decision I make, but I like your phrasing better.

• Actually, we don’t know that our decision affects the contents of Box B. In fact, we’re told that it contains a million dollars if-and-only-if Omega predicts we will only take Box B.

It is possible that we could pick Box B even tho Omega predicted we would take both boxes. Omega has only observed to have predicted correctly 100 times. And if we are sufficiently doubtful whether Omega would predict that we would take only Box B, it would be rational to take both boxes.

Only if we’re somewhat confident of Omega’s prediction can we confidently one-box and rationally expect it to contain a million dollars.

• somewhat confident of Omega’s prediction

51% confidence would suffice.

• Two-box expected value: 0.51 \$1K + 0.49 \$1.001M = \$491000

• One-box expected value: 0.51 \$1M + 0.49 \$0 = \$510000

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

• Newcomb’s problem doesn’t specify how Omega chooses the ‘customers’. It’s a quite realistic possibility that it simply has not offered the choice to anyone that would use a randomizer, and cherrypicked only the people which have at least 99.9% ‘prediction strength’.

• (And the most favourable plausible outcome for randomizing would be scaling the payoff appropriately to the probability assigned.)

• Would that make you a supersuperintelligence? Since I presume by “picking randomly” you mean randomly to Omega, in other words Omega cannot find and process enough information to predict you well.

Otherwise what does “picking randomly” mean?

• The definition of omega as something that can predict your actions leads it to have some weird powers. You could pick a box based on the outcome of a quantum event with a 50% chance, then omega would have to vanish in a puff of physical implausibility.

• I suspect Omega would know you were going to do that, and would be able to put the box in a superposition dependent on the same quantum event, so that in the branches where you 1-box, box B contains \$1million, and where you 2-box it’s empty.

• What’s wrong with Omega predicting a “quantum event”? “50% chance” is not an objective statement, and it may well be that Omega can predict quantum events. (If not, can you explain why not, or refer me to an explanation?)

• From wikipedia

“In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space.[9] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments.”

This is the best formalism we have for predicting things at this scale and it only spits out probabilities. I would be surprised if something did a lot better!

• As I understand it, probabilities are observed because there are observers in two different amplitude blobs of configuration space (to use the language of the quantum physics sequence) but “the one we are in” appears to be random to us. And mathematically I think quantum mechanics is the same under this view in which there is no “inherent, physical” randomness (so it would still be the best formalism we have for predicting things).

Could you say what “physical randomness” could be if we don’t allow reference to quantum mechanics? (i.e. is that the only example? and more to the point, does the notion make any sense?)

• You seem to have transitioned to another argument here… please clarify what this has to do with omega and its ability to predict your actions.

• The new argument is about whether there might be inherently unpredictable things. If not, then your picking a box based on the outcome of a “quantum event” shouldn’t make Omega any less physically plausible,

• What I didn’t understand is why you removed quantum experiments from the discussion. I believe it is very plausible to have something that is physically unpredictable, as long as the thing doing the predicting is bound by the same laws as what you are trying to predict.

Consider a world made of reversible binary gates with the same number of inputs as outputs (that is every input has a unique output, and vice versa).

We want to predict one complex gate. Not a problem, just clone all the inputs and copy the gate. However you have to do that only using reversible binary gates. Lets start with cloning the bits.

In is what you are trying to copy without modifying so that you can predict what affect it will have on the rest of the system. You need a minimum of two outputs, so you need another input B.

You get to create the gate in order to copy the bit and predict the system. The ideal truth table looks something like

`In | B | Out | Copy`

`0 | 0 | 0 | 0`

`0 | 1 | 0 | 0`

`1 | 0 | 1 | 1`

`1 | 1 | 1 | 1`

This violates our reversibility assumption. The best copier we could make is

`In | B | Out | Copy`

`0 | 0 | 0 | 0`

`0 | 1 | 1 | 0`

`1 | 0 | 0 | 1`

`1 | 1 | 1 | 1`

This copies precisely, but mucks up the output making our copy useless for prediction. If you could control B, or knew the value of B then we could correct the Output. But as I have shown here finding out the value of a bit is non-trivial. The best we could do would be to find sources of bits with statistically predictable properties then use them for duplicating other bits.

The world is expected to be reversible, and the no cloning theorem applies to reality which I think is stricter than my example. However I hope I have shown how a simple lawful universe can be hard to predict by something inside it.

In short, stop thinking of yourself (and Omega) as an observer outside physics that does not interact with the world. Copying is disturbing.

• I believe it is very plausible to have something that is physically unpredictable, as long as the thing doing the predicting is bound by the same laws as what you are trying to predict.

[attempted proof omitted]

I hope I have shown how a simple lawful universe can be hard to predict by something inside it.

In short, stop thinking of yourself (and Omega) as an observer outside physics that does not interact with the world. Copying is disturbing.

Even though I do not have time to reflect on the attempted proof and even though the attempted proof is best described as a stab at a sketch of a proof and even though this “reversible logic gates” approach to a proof probably cannot be turned into an actual proof and even though Nick Tarleton just explained why the “one box or two box depending on an inherently unpredictable event” strategy is not particularly relevant to Newcomb’s, I voted this up and I congratulate the author (whpearson) because it is an attempt at an original proof of something very cool (namely, limits to an agent’s ability to learn about its environment) and IMHO probably relevant to the Friendliness project. More proofs and informed stabs at proofs, please!

• 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.

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

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

## Suckah!

Point: arguments don’t matter, winning does.

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

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

Yes.

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

Yes.

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

No.

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

No.

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?

Yes.

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

No, it can’t. (But it already did.)

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

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

It seems that my questions were more useful than yours. Perhaps Joyce beffudled you? It could be that he missed something. (Apart from counter-factual \$9,999,000.)

I responded to all your questions with the answers you intended to make the point that I don’t believe those responses are at all incompatible with making the decision that earns you lots and lots of money.

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

Yes.

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

Yes.

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

No.

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

No.

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?

Yes.

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

No, it can’t. (But it already did.)

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

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

It seems that my questions were more useful than yours. Perhaps Joyce beffudled you? It could be that he missed something. (Apart from counter-factual \$9,999,000.)

I responded to all your questions with the answers you intended to make the point that I don’t believe those responses are at all incompatible with making the decision that earns you lots and lots of money.

• To quote E.T. Jaynes:

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

• @: Hal Finney:

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

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

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

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

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

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

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

• Prediction <-> our choice, if we use the 100100 record as equivalent with complete predictive accuracy.

The “weird thing going on here” is that one value is set (that’s what “he has already flown away” does), yet we are being told that we can change the other value. You see these reactions:

1) No, we can’t toggle the other value, actually. Choice is not really in the premise, or is breaking the premise.

2) We can toggle the choice value, and it will set the predictive value accordingly. The prior value of the prediction does not exist or is not relevant.

We have already equated “B wins” with “prediction value = B” wlog. If we furthermore have equated “choice value = B” with “prediction value = B” wlog, we have two permissible arrays of values: all A, or all B. Now our knowledge is restricted to choice value. We can choose A or B. Since the “hidden” values are known to be identical to the visible value, we should pick the visible value in accordance with what we want for a given other value.

Other thoughts:

-Locally, it appears that you cannot “miss out” because within a value set, your choice value is the only possible one in identity with the other values.

-This is a strange problem, because generally paradox provokes these kinds of responses. In this case, however, fixing a value does not cause a contradiction both ways. If you accept the premise and my premises above, there should be no threat of complications from Omega or anything else.

-if 1 and 2 really are the only reactions, and 2 ->onebox, any twoboxers must believe 1. But this is absurd. So whence the twoboxers?

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

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

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

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

If a), then practical reason is 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 100100 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 56 or 45 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.

• 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%

If this was the only chance you ever get to determine your lifespan—then choose B.

In the real world, it would probably be a better idea to discard both options and use your natural lifespan to search for alternative paths to immortality.

• I disagree, not surprisingly, since I was the author of the comment to which you are responding. I would choose A, and I think anyone sensible would choose A. There’s not much one can say here in the way of argument, but it is obvious to me that choosing B here is following your ideals off a cliff. Especially since I can add a few hundred 9s there, and by your argument you should still choose B.

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

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

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

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

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

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

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

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

A = Contents of box A

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

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

• A very good point. I’m the type to stay home from the polls. But I’d also one-box..… hm.

I think it may have to do with the very weak correlation between my choice to vote and the choice of those of a similar mind to me to vote as opposed to the very strong correlation between my choice to one-box and Omega’s choice to put \$1,000,000 in box B.

• Rational agents defect against a bunch of irrational fools who are mostly choosing for signalling purposes and who may well vote for the other guy even if they cooperate.

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

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

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

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

• Um, AIXI is not computable. Relatedly, K(AIXI) is undefined, as AIXI is not a finite object.

Also, A can simulate B, even when K(B)>K(A). For example, one could easily define a computer program which, given sufficient computing resources, simulates all Turing machines on all inputs. This must obviously include those with much higher Kolmogorov complexity.

Yes, you run into issues of two Turing machines/​agents/​whatever simulating each other. (You could also get this from the recursion theorem.) What happens then? Simple: neither simulation ever halts.

• But a rational choice by the agent requires that it simulate Omega

Not so. I don’t need to simulate a hungry tiger in order to stay safely (and rationally) away from it, even though I don’t know the exact methods by which its brain will identify me as a tasty treat. If you think that one can’t “rationally” stay away from hungry tigers, then we’re using the word “rationally” vastly differently.

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

OR

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.

• The key point you’ve missed in your analysis, however, is that Omega is almost always correct in his predictions.

It doesn’t matter how Omega does it—that is a separate problem. You don’t have enough information about his process of prediction to make any rational judgment about it except for the fact that it is a very, very good process. Brain scans, reversed causality, time travel, none of those ideas matter. In the paradox as originally posed, all you have are guesses about how he may have done it, and you would be an utter fool to give higher weight to those guesses than to the fact that Omega is always right.

The if observations (that Omega is always right) disagree with theory (that Omega cannot possibly be right), it is the theory that is wrong, every time.

Thus the rational agent should, in this situation, give extremely low weight to his understanding of the way the universe works, since it is obviously flawed (the existence of a perfect predictor proves this). The question really comes down to 100% chance of getting \$1000 plus a nearly 0% chance of getting \$1.01 million, vs nearly 100% chance of getting \$1 million.

What really blows my mind about making the 2-box choice is that you can significantly reduce Omega’s ability to predict the outcome, and unless you are absolutely desperate for that \$1000* the 2-box choice doesn’t become superior until Omega is only roughly 50% accurate (at 50.1% the outcome equalizes). Only then do you expect to get more money, on average, by choosing both boxes.

In other words, if you think Omega is doing anything but flipping a coin to determine the contents of box B, you are better off choosing box B.

*I could see the value of \$1000 rising significantly if, for example, a man is holding a gun to your head and will kill you in two minutes if you don’t give him \$1000. In this case, any uncertainty of Omega’s abilities are overshadowed by the certainty of the \$1000. This inverts if the man with the gun is demanding more than \$1000 - making the 2-box choice a non-option.

• 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.

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

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

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

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

• Hi. This is a rather old post, so you might not get too many replies.

Newcomb’s problem often comes with the caveat that, if Omega thinks you’re going to game the system, it will leave you with only the \$1,000. But yes, we like clever answers here, although we also like to consider, for the purposes of thought experiments, the least convenient possible world in which the loopholes we find have been closed.

Also, may I suggest visiting the welcome thread?

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

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

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

a) with probability p there is 1M in one box, and 1k 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?

• “Would you really be willing to give up .8 probability of 3^^^3 years of life for a 1/​3^^^3 chance at a longer (but still finite) one?”

I’d like to hear this too.

• Okay. There’s two intuitive obstacles, my heuristic as a human that my mind is too weak to handle tiny probabilities and that I should try to live my life on the mainline, and the fact that 3^^^3 already extrapolates a mind larger than the sum of every future experience my present self can empathize with.

But I strongly suspect that answering “No” would enable someone to demonstrate circular /​ inconsistent preferences on my part, and so I very strongly suspect that my reflective equilibrium would answer “Yes”. Even in the realm of the computable, there are simple computable functions that grow a heck of a lot faster than up-arrow notation.

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

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

So, is it a bet?

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

• So, is it a bet?

No for two reasons—first, I don’t trust human reason including my own when trying to live one’s life inside tiny probabilities of huge payoffs; second, I ordinarily consider myself an average utilitarian and I’m not sure this is how my average utilitarianism plays out. It’s one matter if you’re working within a single universe in which all-but-infinitesimal of the value is to be found within those lives that are infinite, but I’m not sure I would compare two differently-sized possible Realities the same way. I am not sure I am willing to say that a finite life weighs nothing in my utility function if an infinite life seems possible—though if both were known to coexist in the same universe, I might have to bite that bullet. (At the opposite extreme, a Bostromian parliament might assign both cases representative weight proportional to probability and let them negotiate the wise action.)

Also I have severe doubts about infinite ethics, but that’s easily fixed using a really large finite number instead (pay everything if time < googolplex, keep \$1 if time > TREE(100), return \$1 later if time between those two bounds).

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

Keep growing the lifespan by huge computational factors, keep slicing near-infinitesimally tiny increments off the probability. (Is there an analogous inconsistency to which I expose myself by answering “No” to the bet above, from trying to treat alternative universes differently than side-by-side spatial reasons?)

• It’s one matter if you’re working within a single universe in which all of the value is to be found within those lives that are infinite, but I’m not sure I would compare two differently-sized Realities the same way. I am not sure I am willing to say that a finite life weighs nothing in my utility function if an infinite life seems possible.

In that case, it’s not that your utility function is unbounded in years lived, but rather your utility for each year lived is a decreasing function of the lifetime of the universe (or perhaps total lifetime of everyone in the universe).

I’ll have to think if that makes sense.

• It’s possible that I’m reasoning as if my utility function is over “fractions of total achievable value” within any given universe. I am not sure if there are any problems with this, even if it’s true.

• That does have quite a bit of intuitive appeal! However, when you look at a possible universe from the outside, there are no levers nor knobs you can turn, and all the value achieved by the time of heat death was already inherent in the configurations right after the big bang--

--so if you do not want “fraction of total achievable value” to be identically one for every possible universe, the definition of your utility function seems to get intertwined with how exactly you divvy up the world into “causal nodes” and “causal arrows”, in a way that does not seem to happen if you define it in terms of properties of the outcome, like how many fulfilling lifes lived. (Of course, being more complicated doesn’t imply being wrong, but it seems worth noting.)

And yes, I’m taking a timeful view for vividness of imagination, but I do not think the argument changes much if you don’t do that; the point is that it seems like number-of-fulfilling-lifes utility can be computed given only the universal wavefunction as input, whereas for fraction-of-achievable-fulfilling-lifes, knowing the actual wavefunction isn’t enough.

Could your proposal lead to conflicts between altruists who have the same values (e.g. number of fulfilling lifes), but different power to influence the world (and thus different total achievable value)?

• After thinking about it, that doesn’t make sense either. Suppose Omega comes to you and says that among the universes that you live in, there is a small fraction that will end in 5 years. He offers to kill you now in those universes, in exchange for granting you a googleplex years of additional life in a similar fraction of universes with time > TREE(100) and where you would have died in less than googleplex years without his help (and where others manage to live to TREE(100) years old if that makes any difference). Would you refuse?

• No. But here, by specification, you’re making all the universes real and hence part of a larger Reality, rather than probabilities of which only a single one is real.

If there were only one Reality, and there were small probabilities of it being due to end in 5 years, or in a googolplex years, and the two cases seemed of equal probability, and Omega offered to destroy reality now if it were only fated to last 5 years, in exchange for extending its life to TREE(100) if it were otherwise fated to last a googolplex years… well, this Reality is already known to have lasted a few billion years, and through, say, around 2 trillion life-years, so if it is due to last only another 5 years the remaining 30 billion life-years are not such a high fraction of its total value to be lost—we aren’t likely to do so much more in just another 5 years, if that’s our limit; it seems unlikely that we’d get FAI in that time. I’d probably still take the offer. But I wouldn’t leap at it.

• In that case, would you accept my original bet if I rephrase it as making all the universes part of a larger Reality? That is, if in the future we have reason to believe that Tegmark’s Level 4 Multiverse is true, and find ourselves living in a universe with time < googolplex, then you’d give you all your assets and future earnings, in return for \$1 of my money if we find ourselves living in a universe with time > TREE(100).

• I wouldn’t, but my reflective equilibrium might very well do so.

I wouldn’t due to willpower failure exceeding benefit of \$1 if I believe my mainline probability is doomed to eternal poverty.

Reflective equilibrium probably would, presuming there’s a substantial probability of >TREE(100), or that as a limiting process the “tiny” probability falls off more slowly than the “long-lived” universe part increases. On pain of inconsistency when you raise the lifespan by large computational factors each time, and slice tiny increments off the probability each time.

• Ok, as long as your utility function isn’t actually unbounded, here’s what I think makes more sense, assuming a Level 4 Multiverse. It’s also a kind of “fractions of total achievable value”.

Each mathematical structure representing a universe has a measure, which represents it’s “fraction of all math”. (Perhaps it’s measure is exponential in zero minus the length of its definition in a formal set theory.) My utility over that structure is bounded by this measure. In other words, if that structure represents my idea of total utopia, when my utility for it would be its measure. If it’s total dystopia, my utility for it would be 0.

Within a universe, different substructures (for example branches or slices of time) also have different measures, and if I value such substructures independently, my utilities for them are also bounded by their measures. For example, in a universe that ends at t = TREE(100), a time slice with t < googolplex has a much higher measure than a random time slice (since it takes more bits to represent a random t).

If I value each person independently (and altruistically), then it’s like average utilitarianism, except each person is given a weight equal to its measure instead of 1/​population.

This proposal has its own counter-intuitive implications, but overall I think it’s better than the alternatives. It fits in nicely with MWI. It also manages to avoid running into problems with infinities.

• For example, in a universe that ends at t = TREE(100), a time slice with t < googolplex has a much higher measure than a random time slice (since it takes more bits to represent a random t).

I have to say this strikes me as a really odd proposal, though it’s certainly interesting from the perspective of the Doomsday Argument if advanced civilizations have a thermodynamic incentive to wait until nearly the end of the universe before using their hoarded negentropy.

But for me it’s hard to see why “reality-fluid” (the name I give your “measure”, to remind myself that I don’t understand it at all) should dovetail so neatly with the information needed to locate events in universes or universes in Level IV. It’s clear why an epistemic prior is phrased this way—but why should reality-fluid behave likewise? Shades of either Mind Projection Fallacy or a very strange and very convenient coincidence.

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

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

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

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

• But for me it’s hard to see why “reality-fluid” (the name I give your “measure”, to remind myself that I don’t understand it at all) should dovetail so neatly with the information needed to locate events in universes or universes in Level IV.

Well, why would it be easier to locate some events or universes than others, unless they have more reality-fluid?

It’s clear why an epistemic prior is phrased this way—but why should reality-fluid behave likewise? Shades of either Mind Projection Fallacy or a very strange and very convenient coincidence.

Why is it possible to describe one mathematical structure more concisely than another, or to specify one computation using less bits than another? Is that just a property of the mind that’s thinking about these structures and computations, or is it actually a property of Reality? The latter seems more likely to me, given results in algorithmic information theory. (I don’t know if similar theorems has been or can be proven about set theory, that the shortest description lengths in different formalizations can’t be too far apart, but it seems plausible.)

Also, recall that in UDT, there is no epistemic prior. So, the only way to get an effect similar to EDT/​CDT w/​ universal prior, is with a weighting scheme over universes/​events like I described.

• I can sort of buy the part where simple universes have more reality-fluid, though frankly the whole setup strikes me as a mysterious answer to a mysterious question.

But the part where later events have less reality-fluid within a single universe, just because they take more info to locate—that part in particular seems really suspicious. MPF-ish.

• I’m far from satisfied with the answer myself, but it’s the best I’ve got so far. :)

• Consider the case where you are trying to value (a) just yourself versus (b) the set of all future yous that satisfy the constraint of not going into negative utility.

The shannon information of the set (b) could be (probably would be) lower than that of (a). To see this, note that the complexity (information) of the set of all future yous is just the info required to specify (you,now) (because to compute the time evolution of the set, you just need the initial condition), whereas the complexity (information) of just you is a series of snapshots (you, now), (you, 1 microsecond from now), … . This is like the difference between a JPEG and an MPEG. The complexity of the constraint probably won’t make up for this.

If the constraint of going into negative utility is particularly complex, one could pick a simple subset of nonnegative utility future yous, for example by specifying relatively simple constraints that ensure that the vast majority of yous satisfying those constraints don’t go into negative utility.

This is problematic because it means that you would assign less value to a large set of happy future yous than to just one future you. A large and exhaustive set of future happy yous is less complex (easier to specify) than just one.

• it’s certainly interesting from the perspective of the Doomsday Argument if advanced civilizations have a thermodynamic incentive to wait until nearly the end of the universe before using their hoarded negentropy

Related: That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi’s paradox (https://​​arxiv.org/​​pdf/​​1705.03394.pdf)

• This looks pretty plausible to me, because it does seem there is some disutility to the simple fact of dying, regardless of how far in the future that happens. So U(live N years) always contains that disutility, whereas U(live forever) does not.

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

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

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

• The question is how to create a formal decision algorithm that will be able to understand the problem and give the right answer (without failing on other such tests). Of course you can solve it correctly if you are not yet poisoned by too much presumptuous philosophy.

• I guess I’m missing something obvious. The problem seems very simple, even for an AI.

The way the problem is usually defined (omega really is omniscient, he’s not fooling you around, etc.) there are only two solutions:

• You take the two boxes, and Omega had already predicted that, meaning that there is nothing in Box B—you win 1000\$

• You take box B only, and Omega had already predicted that, meaning that there is 1M\$ in box B—you win 1M\$.

That’s it. Period. Nothing else. Nada. Rien. Nichts. Sod all. These are the only two possible options (again, assuming the hypotheses are true). The decision to take box B only is a simple outcome comparison. It is a perfectly rational decision (if you accept the premises of the game).

Now the way Eliezer states it is different from the usual formulation. In Eliezer’s version, you cannot be sure about Omega’s absolute accuracy. All you know is his previous record. That does complicate things, if only because you might be the victim of a scam (e.g. like the well-known trick to convince comeone that you can consistently predict the winning horse in a 2-horse race—simply start with 2^N people, always give a different prediction to each half of them, discard those to whom you gave the wrong one, etc.)

At any rate, the other two outcomes that were impossible in the previous version (involving mis-prediction by Omega) are now possible, with a certain probability that you need to somehow ascertain. That may be difficult, but I don’t see any logical paradox.

For example, if this happened in the real world, you might reason that the probability that you are being scammed is overwhelming in regard to the probability of existence of a truly omniscient predictor. This is a reasonable inference from the fact that we hear about scams every day, but nobody has ever reported such an omniscient predictor. So you would take both boxes and enjoy your expected \$1000+epsilon (Omega may have been sincere but deluded, lucky in the previous 100 trials, and wrong in this one).

In the end, the guy who would win most (in expected value!) would not be the “least rational”, but simply the one who made the best estimates for the probabilites of each outcome, based on his own knowledge of the universe (if you have a direct phone line to the Angel Gabriel, you will clearly do better).

What is the part that would be conceptually (as opposed to technically/​practically) difficult for an algorithm?

• 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.

• What wedrifid said. See also Rationality is Systematized Winning and the section of What Do We Mean By “Rationality”? about “Instrumental Rationality”, which is generally what we mean here when we talk about actions being rational or irrational. If you want to get more money, than the instrumentally rational action is the epistemically rational answer to the question “What course of action will cause me to get the most money?”.

If you accept the premises of Omega thought experiments, then the right answer is one-boxing, period. If you don’t accept the premises, it doesn’t make sense for you to be answering it one way or the other.

• I thought about this last night and also came to the conclusion that randomizing my choice would not “assume the worst” as I ought to.

And I fully accept that this is just a thought experiment & physics is a cheap way out. I will now take the premises or leave them. :)

• It is a common assumption in these sorts of problems that if Omega predicts that you will condition your choice on a quantum event, it will not put the money in Box B.

• Of course two boxes is the rational solution—it’s not as if post-facto cogitation is going to change anything.

No it isn’t. If you like money it is rational to get more money. Take one box.

• I suspect it’s the idea that my choice is totally predictable

At face, that does sound absurd. The problem is that you are underestimating a superintelligence. Imagine that the universe is a computer simulation, so that a set of physical laws plus a very, very long string of random numbers is a complete causal model of reality. The superintelligence knows the laws and all of the random numbers. You still make a choice, even though that choice ultimately depends on everything that preceded it. See http://​​wiki.lesswrong.com/​​wiki/​​Free_will

I think much of the debate about Newcomb’s Problem is about the definition of superintelligence.

• 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!

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

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

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

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

• Sometime ago I figured out a refutation of this kind of reasoning in Counterfactual Mugging, and it seems to apply in Newcomb’s Problem too. It goes as follows:

Imagine another god, Upsilon, that offers you a similar two-box setup—except to get the \$2M in the box B, you must be a one-boxer with regard to Upsilon and a two-boxer with regard to Omega. (Upsilon predicts your counterfactual behavior if you’d met Omega instead.) Now you must choose your dispositions wisely because you can’t win money from both gods. The right disposition depends on your priors for encountering Omega or Upsilon, which is a “bead jar guess” because both gods are very improbable. In other words, to win in such problems, you can’t just look at each problem individually as it arises—you need to have the correct prior/​predisposition over all possible predictors of your actions, before you actually meet any of them. Obtaining such a prior is difficult, so I don’t really know what I’m predisposed to do in Newcomb’s Problem if I’m faced with it someday.

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

On the other hand, beating Newcomb’s Problem is easy, once you let go of the idea that to be “rational” means performing a strange ritual cognition in which you must only choose on the basis of physical consequences and not on the basis of correct predictions that other agents reliably make about you, so that (if you choose using this bizarre ritual) you go around regretting how terribly “rational” you are because of the correct predictions that others make about you. I simply choose on the basis of the correct predictions that others make about me, and so I do not regret being rational.

And these questions are highly relevant and realistic, unlike Upsilon; in the future we can expect there to be lots of rational agents that make good predictions about each other.

• Omega lets me decide to take only one box after meeting Omega, when I have already updated on the fact that Omega exists, and so I have much better knowledge about which sort of god I’m likely to encounter.

In what sense can you update? Updating is about following a plan, not about deciding on a plan. You already know that it’s possible to observe anything, you don’t learn anything new about environment by observing any given thing. There could be a deep connection between updating and logical uncertainty that makes it a good plan to update, but it’s not obvious what it is.

• Huh? Updating is just about updating your map. (?) The next sentence I didn’t understand the reasoning of, could you expand?

• Intuitively, the notion of updating a map of fixed reality makes sense, but in the context of decision-making, formalization in full generality proves elusive, even unnecessary, so far.

By making a choice, you control the truth value of certain statements—statements about your decision-making algorithm and about mathematical objects depending on your algorithm. Only some of these mathematical objects are part of the “real world”. Observations affect what choices you make (“updating is about following a plan”), but you must have decided beforehand what consequences you want to establish (“[updating is] not about deciding on a plan”). You could have decided beforehand to care only about mathematical structures that are “real”, but what characterizes those structures apart from the fact that you care about them?

• Pascal’s Wagers, huh. So your decision theory requires a specific prior?

• This is not a refutation, because what you describe is not about the thought experiment. In the thought experiment, there are no Upsilons, and so nothing to worry about. It is if you face this scenario in real life, where you can’t be given guarantees about the absence of Upsilons, that your reasoning becomes valid. But it doesn’t refute the reasoning about the thought experiment where it’s postulated that there are no Upsilons.

• Thanks for dropping the links here. FWIW, I agree with your objection. But at the very least, the people claiming they’re “one-boxers” should also make the distinction you make.

Also, user Nisan tried to argue that various Upsilons and other fauna must balance themselves out if we use the universal prior. We eventually took this argument to email, but failed to move each other’s positions.

• Just didn’t want you confusing people or misrepresenting my opinion, so made everything clear. :-)

• OK. I assume the usual (Omega and Upsilon are both reliable and sincere, I can reliably distinguish one from the other, etc.)

Then I can’t see how the game doesn’t reduce to standard Newcomb, modulo a simple probability calculation, mostly based on “when I encounter one of them, what’s my probability of meeting the other during my lifetime?” (plus various “actuarial” calculations).

If I have no information about the probability of encountering either, then my decision may be incorrect—but there’s nothing paradoxical or surprising about this, it’s just a normal, “boring” example of an incomplete information problem.

you need to have the correct prior/​predisposition over all possible predictors of your actions, before you actually meet any of them.

I can’t see why that is—again, assuming that the full problem is explained to you on encountering either Upsilon or Omega, both are truhful, etc. Why can I not perform the appropriate calculations and make an expectation-maximising decision even after Upsilon-Omega has left? Surely Omega-Upsilon can predict that I’m going to do just that and act accordingly, right?

• Yes, this is a standard incomplete information problem. Yes, you can do the calculations at any convenient time, not necessarily before meeting Omega. (These calculations can’t use the information that Omega exists, though.) No, it isn’t quite as simple as you state: when you meet Omega, you have to calculate the counterfactual probability of you having met Upsilon instead, and so on.

• I’m pretty sure the logic is correct. I do make silly math mistakes sometimes, but I’ve tested this one on Vladimir Nesov and he agrees. No comment from Eliezer yet (this scenario was first posted to decision-theory-workshop).

• It reminds me vaguely of Pascal’s Wager, but my cached responses thereunto are not translating informatively.

• Then I think the original Newcomb’s Problem should remind you of Pascal’s Wager just as much, and my scenario should be analogous to the refutation thereof. (Thereunto? :-)

• But wait! You can fix this. All you have to do is be disposed to take just Box B.

No, that’s not what I should do. What I should do is make Omega think that I am disposed to take just Box B. If I can successfully make Omega think that I’ll take only Box B but still take both boxes, then I should. But since Omega is superintelligent, let’s take it as understood that the only way to make Omega think that I’ll take only Box B is to make it so that I’ll actually take Box B. Then that is what I should do.

But I have to do it now! (I don’t do it now only because I don’t believe that this situation will ever happen.) Once Omega has placed the boxes and left, if the known laws of physics apply, then it’s too late!

If you take only Box B and get a million dollars, wouldn’t you regret having not also taken Box A? Not only would you have gotten a thousand dollars more, you’d also have shown up that know-it-all superintelligent intergalactic traveller too! That’s a chance that I’ll never have, since Omega will read my comment here and leave my Box B empty, but you might have that chance, and if so then I hope you’ll take it.

• It’s not really too late then. Omega can predict what you’ll do between seeing the boxes, and choosing which to take. If this is going to include a decision to take one box, then Omega will put a million dollars in that box.

I will not regret taking only one box. It strikes me as inconsistent to regret acting as the person I most wish to be, and it seems clear that the person I most wish to be will take only one box; there is no room for approved regret.

• It’s not really too late then.

If you say this, then you believe in backwards causality (or a breakdown of the very notion of causality, as in Kevin’s comment below). I agree that if causality doesn’t work, then I should take only Box B, but nothing in the problem as I understand it from the original post implies any violation of the known laws of physics.

If known physics applies, then Omega can predict all it likes, but my actions after it has placed the boxes cannot affect that prediction. There is always the chance that it predicts that I will take both boxes but I take only Box B. There is even the chance that it will predict that I will take only Box B but I take both boxes. Nothing in the problem statement rules that out. It would be different if that were actually impossible for some reason.

I will not regret taking only one box.

I knew that you wouldn’t, of course, since you’re a one-boxer. And we two-boxers will not regret taking both boxes, even if we find Box B empty. Better \$1000 than nothing, we will think!

• If known physics applies, then Omega can predict all it likes, but my actions after it has placed the boxes cannot affect that prediction. There is always the chance that it predicts that I will take both boxes but I take only Box B. There is even the chance that it will predict that I will take only Box B but I take both boxes. Nothing in the problem statement rules that out. It would be different if that were actually impossible for some reason.

Ah, I see what the probem is. You have a confused notion of free will and what it means to make a choice.

Making a choice between two options doesn’t mean there is a real chance that you might take either option (there always is at least an infinitesimal chance, but that it always true even for things that are not usefully described as a choice). It just means that attributing the reason for your taking whatever option you take is most usefully attributed to you (and not e.g. gravity, government, the person holding a gun to you head etc.). In the end, though, it is (unless the choice is so close that random noise makes the difference) a fact about you that you will make the choice you will make. And it is in principle possible for others to discover this fact about you.

If it is a fact about you that you will one-box it is not possible that you will two-box. If it is a fact about you that you will two-box it is not possible that you will one-box. If it is a fact about you that you will leave the choice up to chance then Omega probably doesn’t offer you to take part in the first place.

Now, when deciding what choice to make it is usually most useful to pretend there is a real possibility of taking either option, since that generally causes facts about you that are more benefitial to you. And that you do that is just another fact about you, and influences the fact about which choice you make. Usually the fact which choice you will make has no consequences before you make your choice, and so you can model the rest of the world as being the same in either case up to that point when counterfactually considering the consequences of either choice. But the fact about which choice you will make is just another fact like any other, and is allowed, even if it usually doesn’t, to have consequences before that point in time. If it does it is best, for the very same reason you pretend that either choice is a real possibility in the first place, to also model the rest of the world as different contingent on your choice. That doesn’t mean backwards causality. Modeling the word in this way is just another fact about you that generates good outcomes.

• Alicorn:

It’s not really too late then. Omega can predict what you’ll do between seeing the boxes, and choosing which to take. If this is going to include a decision to take one box, then Omega will put a million dollars in that box.

TobyBartels:

If you say this, then you believe in backwards causality (or a breakdown of the very notion of causality, as in Kevin’s comment below). I agree that if causality doesn’t work, then I should take only Box B, but nothing in the problem as I understand it from the original post implies any violation of the known laws of physics.

I remember reading an article about someone who sincerely lacked respect for people who were ‘soft’ (not exact quote) on the death penalty … before ending up on the jury of a death penalty case, and ultimately supporting life in prison instead. It is not inconceivable that a sufficiently canny analyst (e.g. Omega) could deduce that the process of being picked would motivate you to reconsider your stance. (Or, perhaps more likely, motivate a professed one-boxer like me to reconsider mine.)

• If you say this, then you believe in backwards causality (or a breakdown of the very notion of causality, as in Kevin’s comment below). I agree that if causality doesn’t work, then I should take only Box B, but nothing in the problem as I understand it from the original post implies any violation of the known laws of physics.

Beware hidden inferences. Taboo causality.

• I don’t see what that link has to do with anything in my comment thread. (I haven’t read most of the other threads in reply to this post.)

I should explain what I mean by ‘causality’. I do not mean some metaphysical necessity, whereby every event (called an ‘effect’) is determined (or at least influenced in some asymmetric way) by other events (called its ‘causes’), which must be (or at least so far seem to be) prior to the effect in time, leading to infinite regress (apparently back to the Big Bang, which is somehow an exception). I do not mean anything that Aristotle knew enough physics to understand in any but the vaguest way.

I mean the flow of macroscopic entropy in a physical system.

The best reference that I know on the arrow of time is Huw Price’s 1996 book Time’s Arrow and Archimedes’ Point. But actually I didn’t understand how entropy flow leads to a physical concept of causality until several years after I read that, so that might not actually help, and I’m having no luck finding the Internet conversation that made it click for me.

But basically, I’m saying that, if known physics applies, then P(there is money in Box B|all information available on a macroscopic level when Omega placed the boxes) = P(there is money in Box B|all information … placed the boxes & I pick both boxes), even though P(I pick both boxes|all information … placed the boxes) < 1, because macroscopic entropy strictly increases between the placing of the boxes and the time that I finally pick a box.

So I need to be given evidence that known physics does not apply before I pick only Box B, and a successful record of predictions by Omega will not do that for me.

• The Psychopath Button: Paul is debating whether to press the ‘kill all psychopaths’ button. It would, he thinks, be much better to live in a world with no psychopaths. Unfortunately, Paul is quite confident that only a psychopath would press such a button. Paul very strongly prefers living in a world with psychopaths to dying. Should Paul press the button? (Set aside your theoretical commitments and put yourself in Paul’s situation. Would you press the button? Would you take yourself to be irrational for not doing so?)

Newcomb’s Firebomb: There are two boxes before you. Box A definitely contains \$1,000,000. Box B definitely contains \$1,000. You have two choices: take only box A (call this one-boxing), or take both boxes (call this two-boxing). You will signal your choice by pressing one of two buttons. There is, as usual, an uncannily reliable predictor on the scene. If the predictor has predicted that you will two-box, he has planted an incendiary bomb in box A, wired to the two-box button, so that pressing the two-box button will cause the bomb to detonate, burning up the \$1,000,000. If the predictor has predicted that you will one-box, no bomb has been planted – nothing untoward will happen, whichever button you press. The predictor, again, is uncannily accurate.

I would suggest looking at your implicit choice of counterfactuals and their role in your decision theory. Standard causal decision theory involves local violations of the laws of physics (you assign probabilities to the world being such that you’ll one-box, or such that you’ll one-box, and then ask what miracle magically altering your decision, without any connection to your psychological dispositions, etc, would deliver the highest utility). Standard causal decision theory is a normative principle for action, that says to do the action that would deliver the most utility if a certain kind of miracle happened. But you can get different versions of causal decision theory by substituting different sorts of miracles, e.g. you can say: “if I one-box, then I have a psychology that one-boxes, and likewise for two-boxing” so you select the action such that a miracle giving you the disposition to do so earlier on would have been better. Yet another sort of counterfactual that can be hooked up to the causal decision theory framework would go “there’s some mathematical fact about what decision(decisions given Everett) my brain structure leads to in standard physics, and the predictor has access to this mathematical info, so I’ll select the action that would be best brought about by a miracle changing that mathematical fact”.

• Thanks for the replies, everybody!

This is a global response to several replies within my little thread here, so I’ve put it at nearly the top level. Hopefully that works out OK.

I’m glad that FAWS brought up the probabilistic version. That’s because the greater the probability that Omega makes mistakes, the more inclined I am to take two boxes. I once read the claim that 70% of people, when told Newcomb’s Paradox in an experiment, claim to choose to take only one box. If this is accurate, then Omega can achieve a 70% level of accuracy by predicting that everybody is a one-boxer. Even if 70% is not accurate, you can still make the paradox work by adjusting the dollar amounts, as long as the bias is great enough that Omega can be confident that it will show up at all in the records of its past predictions. (To be fair, the proportion of two-boxers will probably rise as Omega’s accuracy falls, and changing the stakes should also affect people’s choices; there may not be a fixed point, although I expect that there is.)

If, in addition to the problem as stated (but with only 70% probability of success), I know that Omega always predicts one-boxing, then (hopefully) everybody agrees that I should take both boxes. There needs to some correlation between Omega’s predictions and the actual outcomes, not just a high proportion of past successes.

FAWS also writes:

You yourself claim to know what you would do in the boxing experiment

Actually, I don’t really want to make that claim. Although I’ve written things like ‘I would take both boxes’, I really should have written ‘I should take both boxes’. I’m stating a correct decision, not making a prediction about my actual actions. Right now, I predict about a 70% chance of two-boxing given the situation as stated in the original post, although I’ve never tried to calculate my estimates of probabilities, so who knows what that really means. (H’m, 70% again? Nope, I don’t trust that calibration at all!)

FAWS writes elsewhere:

Making a choice between two options […] just means that attributing the reason for your taking whatever option you take is most usefully attributed to you (and not e.g. gravity, government, the person holding a gun to you head etc.).

I don’t see what the gun has to do with it; this is a perfectly good problem in decision theory:

• Suppose that you have a button that, if pressed, will trigger a bomb that kills two strangers on the other side of the world. I hold a gun to your head and threaten to shoot you if you don’t press the button. Should you press it?

A person who presses the button in that situation can reasonably say afterwards ‘I had no choice! Toby held a gun to my head!’, but that doesn’t invalidate the question. Such a person might even panic and make the question irrelevant, but it’s still a good question.

If it is a fact about you that you will leave the choice up to chance then Omega probably doesn’t offer you to take part in the first place.

So that’s how Omega gets such a good record! (^_^)

Understanding the question really is important. I’ve been interpreting it something along these lines: you interrupt your normal thought processes to go through a complete evaluation of the situation before you, then see what you do. (This is exactly what you cannot do if you panic in the gun problem above.) So perhaps we can predict with certain accuracy that an utter bigot will take one course of action, but that is not what the bigot should do, nor is it what they will do if they discard their prejudices and decide afresh.

Now that I think about it, I see some problems with this interpretation, and also some refinements that might fix it. (The first thing to do is to make it less dependent on the specific person making the decision.) But I’ll skip the refinements. It’s enough to notice that Omega might very well predict that a person will not take the time to think things through, so there is poor correlation between what one should do and what Omega will predict, even though the decision is based on what the world would be like if one did take the time.

I still think that (modulo refinements) this is a good interpretation of what most people would mean if they tell a story and then ask ‘What should this person do?’. (I can try to defend that claim if anybody still wants me to after they finish this comment.) In that case, I stand by my decision that one should take both boxes, at least if there is no good evidence of new physics.

However, I now realise that there is another interpretation, which is more practical, however much the ordinary person might not interpret things this way. That is: sit down and think through the whole situation now, long before you are ever faced with it in real life, and decide what to do. One obvious benefit of this is that when I hold a gun to your head, you won’t panic, because you will be prepared. More generally, this is what we are all actually doing right now! So as we make these idle philosophical musings, let’s be practical, and decide what we’ll do if Omega ever offers us this deal.

In this case, I agree that I will be better off (given the extremely unlikely but possible assumption that I am ever in this situation) if I have decided now to take only Box B. As RobinZ points out, I might change my mind later, but that can’t be helped (and to a certain extent shouldn’t be helped, since it’s best if I take two boxes after Omega predicts that I’ll only take one, but we can’t judge that extent if Omega is smarter than us, so really there’s no benefit to holding back at all).

If Omega is fallible, then the value of one-boxing falls drastically, and even adjusting the amount of money doesn’t help in the end; once Omega’s proportion of past success matches the observed proportion in experiments (or whatever our best guess of the actual proportion of real people is), then I’m back to two-boxing, since I expect that Omega simply always predicts one-boxing.

In hindsight, it’s obvious that the the original post was about decision in this sense, since Eliezer was talking about an AI that modifies its decision procedures in anticipation of facing Omega in the future. Similarly, we humans modify our decision procedures by making commitments and letting ourselves invent rationalisations for them afterwards (although the problem with this is that it makes it hard to change our minds when we receive new information). So obviously Eliezer wants us to decide now (or at least well ahead of time) and use our leet Methods of Rationality to keep the rationalisations in check.

So I hereby decide that I will pick only one box. (You hear that, Omega!?) Since I am honest (and strongly doubt that Omega exists), I’ll add that I may very well change my mind if this ever really happens, but that’s about what I would do, not what I should do. And in a certain sense, I should change my mind … then. But in another sense, I should (and do!) choose to be a one-boxer now.

(Thanks also to CarlShulman, whom I haven’t quoted, but whose comment was a big help in drawing my attention to the different senses of ‘should’, even though I didn’t really adopt his analysis of them.)

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

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

If you choose to two-box:

• X chance of getting \$1000

• (1-X) chance of getting \$1,001,000

If you choose to take box B only:

• X chance of getting \$1,000,000

• (1-X) chance of getting \$0

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

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

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

X < 0.5005

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

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

Your caclulation is fine. What you’re missing is that Omega has a record of 70% accuracy because Omega always predicts that a person will one-box and 70% of people are one-boxers. So Omega always puts the million dollars in Box B, and I will always get \$1,001,000\$ if I’m one of the 30% of people who two-box.

At least, that is a possibility, which your calculation doesn’t take into account. I need evidence of a correlation between Omega’s predictions and the participants’ actual behaviour, not just evidence of correct predictions. My prior probability distribution for how often people one-box isn’t even concentrated very tightly around 70% (which is just a number that I remember reading once as the result of one survey), so anything short of a long run of predictions with very high proportion of correct ones will make me suspect that Omega is pulling a trick like this.

So the problem is much cleaner as Eliezer states it, with a perfect record. (But if even that record is short, I won’t buy it.)

• Oops, I see that RobinZ already replied, and with calculations. This shows that I should still remove the word ‘drastically’ from the bit that nhamann quoted.

• Wait—we can’t assume that the probability of being correct is the same for two-boxing and one-boxing. Suppose Omega has a probability X of predicting one when you choose one and Y of predicting one when you choose two.

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

The special case you list corresponds to Y = 1 - X, but in the general case, we can derive that E1 > E2 implies

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

If we assume linear utility in wealth, this corresponds to a difference of 0.001. If, alternately, we choose a median net wealth of \$93 100 (the U.S. figure) and use log-wealth as the measure of utility, the required difference increases to 0.004 or so. Either way, unless you’re dead broke (e.g. net wealth \$1), you had better be extremely confident that you can fool the interrogator before you two-box.

• You underestimate the meaning of superintelligence. One way of defining a superintelligence that wins at Newcomb without violating causality, is to assume that the universe is computer simulation like, such that it can be defined by a set of physical laws and a very long string of random numbers. If Omega knows the laws and random numbers that define the universe, shouldn’t Omega be able to predict your actions with 100% accuracy? And then wouldn’t you want to choose the action that results in you winning a lot more money?

• So part of the definition of a superintelligence is that the universe is like that and Omega knows all that? In other words, if I have convincing evidence that Omega is superintelligent, then I must have convincing evidence that the universe is a computer simulation, etc? Then that changes things; just as the Second Law of Thermodynamics doesn’t apply to Maxwell’s Demon, so the law of forward causality (which is actually a consequence of the Second Law, under the assumption of no time travel) doesn’t apply to a superintelligence. So yes, then I would pick only Box B.

This just goes to show how important it is to understand exactly what the problem states.

• The computer simulation assumption isn’t necessary, the only thing that matters is that Omega is transcendentally intelligent, and it has all the technology that you might imagine a post-Singularity intelligence might have (we’re talking Shock Level 4). So Omega scans your brain by using some technology that is effectively indistinguishable from magic, and we’re left to assume that it can predict, to a very high degree of accuracy, whether you’re the type of person who would take one or two boxes.

Omega doesn’t have to actually simulate your underlying physics, it just needs a highly accurate model, which seems reasonably easy to achieve for a superintelligence.

• If its model is good enough that it violates the Second Law as we understand it, fine, I’ll pick only Box B, but I don’t see anything in the problem statement that implies this. The only evidence that I’m given is that it’s made a run of perfect predictions (of unknown length!), is smarter than us, and is from very far away. That’s not enough for new physics.

And just having a really good simulation of my brain, of the sort that we could imagine doing using known physics but just don’t have the technical capacity for, is definitely not good enough. That makes the probability that I’ll act as predicted very high, but I’ll still come out worse if, after the boxes have been set, I’m unlucky enough to only pick Box B anyway (or come out better if I’m lucky enough to pick both boxes anyway, if Omega pegs me for a one-boxer).

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

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

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

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

• 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

• For the future, perhaps this once again updated link may help: Updated link

Citation: LEDWIG, Marion, 2000. Newcomb’s problem [Dissertation]. Konstanz: University of Konstanz

• 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?

• One factor you may not have considered: the obvious rational metastrategy is causal decision theory, and causal decision theory picks the two-box strategy.

• I don’t follow. Isn’t it precisely on the meta-strategy level that CDT becomes obviously irrational?

• Key word is “obvious”. If you say, “how should you solve games?”, the historical answer is “using game theory”, and when you say, “what does game theory imply for Newcomb’s dilemma?”, the historical answer is “two-box”. It takes an additional insight to work out that a better metastrategy is possible, and things which take an additional insight are no longer obvious, true or no.

Edit: Alternatively: When I said “metastrategy”, I meant one level higher than “two-boxing”—in other words, the level of decision theory. (I’m not sure which of the two objections you were raising.)

• This is basically what I was trying to point out. :)

• I think what RobinZ means is that you want to choose a strategy such that having that strategy will causally yield nice things. Given that criterion, object-level CDT fails; but one uses a causal consideration to reject it.

• 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.

It is the obvious rational strategy… which is why using a decision theory that doesn’t get this wrong is important.

• Yup yup, you’re right, of course.

What I was trying to say, then, is that I don’t understand why there’s any debate about the validity of a decision theory that gets this wrong. I’m surprised everyone doesn’t just go, “Oh, obviously any decision theory that says two-boxing is ‘rational’ is an invalid theory.”

I’m surprised that this is a point of debate. I’m surprised, so I’m wondering, what am I missing?

Did I manage to make my question clearer like that?

• I can say that for me personally, the hard part—that I did not get past till reading about it here—was noticing that there is actually such a variable as “what decision theory to use”; using a naive CDT sort of thing simply seemed rational /​a priori/​. Insufficient grasp of the nameless virtue, you could say.

• Meaning you’re in the same boat as me? Confused as to why this ever became a point of debate in the first place?

• ...no? I didn’t realize that the decision theory could be varied, that the obvious decision theory could be invalid, so I hit a point of confusion with little idea what to do about it.

• But you’re not saying that you would ever have actually decided to two-box rather than take box B if you found yourself in that situation, are you?

I mean, you would always have decided, if you found yourself in that situation, that you were the kind of person Omega would have predicted to choose box B, right?

I am still so majorly confused here. :P

• I have no idea! IIRC I leaned towards one-boxing, but I was honestly confused about it.

• Ahah. So do you remember if you were confused in yourself, for reasons generated by your own brain, or just by your knowledge that some experts were saying two-boxing was the ‘rational’ strategy?

• It’s a good question. You aren’t missing anything. And “people are crazy, the world is mad” isn’t always sufficient. ;)

• Ha! =]

Okay, I DO expect to see lots of ‘people are crazy, the world is mad’ stuff, yeah, I just wouldn’t expect to see it on something like this from the kind of people who work on things like Causal Decision Theory! :P

So I guess what I really want to do first is CHECK which option is really most popular among such people: two-boxing, or predictably choosing box B?

Problem is, I’m not sure how to perform that check. Can anyone help me there?

• It is fairly hard to perform such checks. We don’t have many situations which are analogous to Newcomb’s problem. We don’t have perfect predictors and most situations humans are in can be considered “iterated”. At least, we can consider most people to be using their ‘iterated’ reasoning by mistake when we put them in once off situations.

The closest analogy that we can get reliable answers out of is the ‘ultimatum game’ with high stakes… in which people really do refuse weeks worth of wages.

By the way, have you considered what you would do if the boxes were transparent? Just sitting there. Omega long gone and you can see piles of cash in front of you… It’s tricky. :)

• Suppose my decision algorithm for the “both boxes are transparent” case is to take only box B if and only if it is empty, and to take both boxes if and only if box B has a million dollars in it. How does Omega respond? No matter how it handles box B, it’s implied prediction will be wrong.

Perhaps just as slippery, what if my algorithm is to take only box B if and only if it contains a million dollars, and to take both boxes if and only if box B is empty? In this case, anything Omega predicts will be accurate, so what prediction does it make?

Come to think of it, I could implement the second algorithm (and maybe the first) if a million dollars weighs enough compared to the boxes. Suppose my decision algorithm outputs: “Grab box B and test it’s weight, and maybe shake it a bit. If it clearly has a million dollars in it, take only box B. Otherwise, take both boxes.” If that’s my algorithm, then I don’t think the problem actually tells us what Omega predicts, and thus what outcome I’m getting.

• The naive presentation of the transparent problem is circular, and for that reason ill defined (what you do depends on what’s in the boxes depends on omega’s prediction depends on what you do...). A plausible version of the transparent newcomb’s problem involves Omega:

1. Predicting what you’d do if you saw box B full (and never mind the case where box B is empty).

2. Predicting what you’d do if you saw box B empty (and never mind the case where box B is full).

3. Predicting what you’d do in both cases, and filling box B if and only if you’d one-box in both of them.

Or variations of those. There’s no circularity when he only makes such “conditional” predictions.

He could use the same algorithms in the non-transparent case, and they would reduce to the normal newcomb’s problem usually, but prevent you from doing any tricky business if you happen to bring an X-ray imager (or kitchen scales) and try to observe the state of box B.

• Suppose my decision algorithm for the “both boxes are transparent” case is to take only box B if and only if it is empty, and to take both boxes if and only if box B has a million dollars in it. How does Omega respond? No matter how it handles box B, it’s implied prediction will be wrong.

Death by lightning.

I typically include such disclaimers such as the above in a footnote or more precisely targeted problem specification so as to avoid any avoid-the-question technicalities. The premise is not that Omega is an idiot or a sloppy game-designer.

Come to think of it, I could implement the second algorithm (and maybe the first) if a million dollars weighs enough compared to the boxes. Suppose my decision algorithm outputs: “Grab box B and test it’s weight, and maybe shake it a bit. If it clearly has a million dollars in it, take only box B. Otherwise, take both boxes.” If that’s my algorithm, then I don’t think the problem actually tells us what Omega predicts, and thus what outcome I’m getting.

You took box B. Putting it down again doesn’t help you. Finding ways to be cleverer than Omega is not a winning solution to Newcomblike problems.

• Suppose my decision algorithm for the “both boxes are transparent” case is to take only box B if and only if it is empty, and to take both boxes if and only if box B has a million dollars in it. How does Omega respond? No matter how it handles box B, it’s implied prediction will be wrong.

• Box B appears full of money; however, after you take both boxes, you find that the money in Box B is Monopoly money. The money in Box A remains genuine, however.

• Box B appears empty, however, on opening it you find, written on the bottom of the box, the full details of a bank account opened by Omega, containing one million dollars, together with written permission for you to access said account.

In short, even with transparent boxes, there’s a number of ways for Omega to lie to you about the contents of Box B, and in this manner control your choice. If Omega is constrained to not lie about the contents of Box B, then it gets a bit trickier; Omega can still maintain an over 90% success rate by presenting the same choice to plenty of other people with an empty box B (since most people will likely take both boxes if they know B is empty).

Or, alternatively, Omega can decide to offer you the choice at a time when Omega predicts you won’t live long enough to make it.

Perhaps just as slippery, what if my algorithm is to take only box B if and only if it contains a million dollars, and to take both boxes if and only if box B is empty? In this case, anything Omega predicts will be accurate, so what prediction does it make?

That depends; instead of making a prediction here, Omega is controlling your choice. Whether you get the million dollars or not in this case depends on whether Omega wants you to have the million dollars or not, in furtherance of whatever other plans Omega is planning.

Omega doesn’t need to predict your choice; in the transparent-box case, Omega needs to predict your decision algorithm.

• “The boxes are transparent” doesn’t literally mean “light waves pass through the boxes” given the description of the problem; it means “you can determine what’s inside the boxes without (and before) opening them”.

Responding by saying “maybe you can see into the boxes but you can’t tell if the money inside is fake” is being hyper-literal and ignoring what people really mean when they specify “suppose the boxes are transparent”.

• Fair enough. I am at times overly literal.

In which case, if you are determined to show that Omega’s prediction is incorrect, and Omega can predict that determination, then the only way that Omega can avoid making an incorrect prediction is either to modify you in some manner (until you are no longer determined to make Omega’s prediction incorrect), or to deny you the chance to make the choice entirely.

For example, Omega might modify you by changing your circumstances; e.g. giving a deadly disease to someone close to you; which can be cured, but only at a total cost of all the money you are able to raise plus \$1000. If Omega then offers the choice (with box B empty) most people would take both boxes, in order to be able to afford the cure.

Alternatively, given such a contrary precommitment, Omega may simply never offer you the choice at all; or might offer you the choice three seconds before you get struck by lightning.

• “Omega puts money inside the boxes, you just never live to get it” is as outside the original problem as “the boxes are transparent, you just don’t understand what you’re seeing when you look in them” is outside the transparent problem. Just because the premise of the problem doesn’t explicitly say ”… and you get the contents of the boxes” doesn’t mean the paradox can be resolved by saying you don’t get the contents of the boxes—that’s being hyper-literal again. Likewise, just because the problem doesn’t say ”… and Omega can’t modify you to change your choice” doesn’t mean that the paradox can be resolved by saying that Omega can modify you.to change your choice—the problem is about decision theory, and Omega doesn’t have capabilities that are irrelevant to what the problem is about.

• The problem, as stated, as far as I can tell gives Omega three options:

• Fail to correctly predict what the person will choose

• Refuse to participate

• Cheat

It is likely that Omega will try to correctly predict what the person will choose; that is, Omega will strive to ignore the first option. If Omega offers the choice to this hypothetical person in the first place, then Omega is not taking the second option.

That leaves the third option; to cheat. I expect that this is the choice that Omega will be most likely to take; one of the easiest ways to do this is by ignoring the spirit of the constraints and taking the exact literal meaning. (Another way is to creatively misunderstand the spirit of the rules as given).

So I provided some suggestions with regard to how Omega might cheat; such as arranging that the decision is never made.

If you think that’s outside the problem, then I’m curious; what do you think Omega would do?

• If you think that’s outside the problem, then I’m curious; what do you think Omega would do?

The point here is that the question is inconsistent. It is impossible for an Omega that can predict with high accuracy to exist, as you’ve correctly pointed out it leads to a situation where Omega must either fail to participate, refuse to participate or cheat, which are all out of bounds of the problem.

• I don’t think it’s ever wise to ignore the possibility of a superintelligent AI cheating, in some manner.

If we ignore that possibility, then yes, the question would be inconsistent; which implies that if the situation were to actually appear to happen, then it would be quite likely that either:

• The situation has been misunderstood; or

• Someone is cheating

Since it is far easier for Omega, being an insane superintelligence, to cheat than it is for someone to cheat Omega, it seems likeliest that if anyone is cheating, then it is Omega.

After all, Omega had and did not take the option to refuse to participate.

• I expect that this is the choice that Omega will be most likely to take; one of the easiest ways to do this is by ignoring the spirit of the constraints and taking the exact literal meaning.

The constraints aren’t constraints on Omega; the constraints are constraints on the reader—they tell the reader what he is supposed to use as the premises of the scenario. Omega cannot cheat unless the reader interprets the description of the problem to mean that Omega is willing to cheat. And if the reader does interpret it that way, it’s the reader, not Omega, who’s violating the spirit of the constraints and being hyper-literal.

what do you think Omega would do?

I think that depending on the human’s intentions, and assuming the human is a perfect reasoner, the conditions of the problem are contradictory. Omega can’t always predict the human—it’s logically impossible.

• In the first case, Omega does not offer you the deal, and you receive \$0, proving that it is possible to do worse than a two-boxer.

In the second case, you are placed into a superposition of taking one box and both boxes, receiving the appropriate reward in each.

In the third case, you are counted as ‘selecting’ both boxes, since it’s hard to convince Omega that grabbing a box doesn’t count as selecting it.

• The premise is that Omega offers you the deal. If Omega’s predictions are always successful because it won’t offer the deal when it can’t predict the result, you can use me as Omega and I’d do as well as him—I just never offer the deal.

The (non-nitpicked version of the) transparent box case shows what’s wrong with the concept: Since your strategy might involve figuring out what Omega would have done, it may be in principle impossible for Omega to predict what you’re going to do, as Omega is indirectly trying to predict itself, leading to an undecideability paradox. The transparent boxes just make this simpler because you can “figure out” what Omega would have done by looking into the transparent boxes.

Of course, if you are not a perfect reasoner, it might be possible that Omega can always predict you, but then the question is no longer “which choice should I make”, it’s “which choice should I make within the limits of my imperfect reasoning”. And answering that requires formalizing exactly how your reasoning is limited, which is rather hard.

• Thanks, but I meant not a check on what these CDT-studying-type people would DO if actually in that situation, but a check on whether they actually say that two-boxing would be the “rational” thing to do in that hypothetical situation.

I haven’t considered you transparency question, no. Does that mean Omega did exactly what he would have done if the boxes were opaque, except that they are in fact transparent (a fact that did not figure into the prediction)? Because in that case I’d just see the million in B, and the thousand in A, and of course take ’em both.

Otherwise, Omega should be able to predict as well as me that, if I knew the rules of this game were that, if I decided to predictably choose to take only box B and leave A alone, box B would contain a million, and both boxes are transparent (and this transparency is figured into the prediction), I would expect to see a million in box B, take it, and just walk away from the paltry thousand in A.

This make sense?

• I think this is the position of classical theorists on self-modifiying agents:

I conclude that the rational action for a player in the Newcomb Paradox is taking both boxes, but that rational agents will usually take only one box because they have rationally adopted the disposition to do so.″

They agree that agents who can self-modify will take one box. But they call that action “irrational”. So, the debate really boils down to the definition of the term “rational”—and is not really concerned with the decision that rational agents who can self-modifiy will actually take.

If my analysis here is correct, the dispute is really all about terminology.

• 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.

• See chapters 1-9 of this document for a more detailed treatment of the argument.

• This link is 404ing. Anyone have a copy of this?

• The current version is here. (It’s Eliezer Yudkowsky (2010). Timeless Decision Theory.)

• 4 Nov 2010 1:32 UTC
4 points

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 :)

• Rather than transforming the problem in the way you did, transform it so that you move first—Omega doesn’t put money in the boxes until you say which one(s) you want.

Given this reframing, f(Always One-Box) seems like a fairly trivial solution, and neither paradoxical nor terribly difficult to represent mathematically...

As a decision problem, Newcomb’s problem is rather pointless, IMHO. As a thought experiment helping us to understand the assumptions that are implicit in game theory, it could be rather useful. The thought experiment shows us that when a problem statement specifies a particular order of moves, what is really being specified is a state of knowledge at decision time. When a problem specifies that Omega moves first that is implicitly in contradiction to the claim that he knows what you will do when you move second. The implicit message is that Omega doesn’t know—the explicit message is that he does. If the explicit message is to be believed, then change the move order to make the implicit message match the explicit one.

However, here, many people seem to prefer to pretend that Newcomb problems constitute a decision theory problem which requires clever solution, rather than a bit of deliberate confusion constructed by violating the implicit rules of the problem genre.

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

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

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

• Let me try my own stab at a little chat with Omega. By the end of the chat I will either have 1001 K, or give up. Right now, I don’t know which.

Act I

Everything happens pretty much as it did in Polymeron’s dialogue, up until…

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.”

Omega: Yup, that’ll work. So you’re happy with your 1000 K?

Act II

Whereupon I try to exploit randomness.

Me: Actually, no. I’m not happy. I want the entire 1001 K. Any suggestions for outsmarting you?

Omega: Nope.

Me: Are you omniscient?

Omega: As far as you’re concerned, yes. Your human physicists might disagree in general, but I’ve got you pretty much measured.

Me: Okay, then. Wanna make a bet? I bet I can find a to get over 1000 K if I make a bet with you. You estimate your probability of being right at 100%, right? Nshepperd had a good suggestion….

Omega: I won’t play this game. Or let you play it with anyone else. I thought we’d moved past that.

Me: How about I flip a fair coin to decide between B and A+B. In fact, I’ll use ’s generator using the principle to generate the outcome of a truly random coin flip. Even you can’t predict the outcome.

Omega: And what do you expect to happen as a result of this (not-as-clever-as-you-think) strategy?

Me: Since you can’t predict what I’ll do, hopefully you’ll fill both boxes. Then there’s a true 50% chance of me getting 1001 K. My expected payoff is 1000.5 K.

Omega: That, of course, is assuming I’ll fill both boxes.

Me: Oh, I’ll make you fill both boxes. I’ll bias the ’s to 50+eps% chance of one-boxing for the expected winnings of 1000.5 K – eps. Then if you want to maximize your omniscience-y-ness, you’ll have to fill both boxes.

Omega: Oh, taking others’ suggestions already? Can’t think for yourself? Making edits to make it look like you’d thought of it in time? Fair enough. Attribute this one to gurgeh. As to the idea itself, I’ll disincentivize you from randomization at all. I won’t fill box B if I predict you cheating.

Me: But then there’s a 50-eps% chance of proving you wrong. I’ll take it. MWAHAHA.

Omega: What an idiot. You’re not trying to prove me wrong. You’re trying to maximize your own profit.

Me: The only reason I don’t insult you back is because I operate under Crackers Rule.

Omega: Crocker’s Rules.

Me: Uh. Right. Whoops.

Omega: Besides. Your ’s random generator idea won’t work even to get you the cheaters’ utility for proving me wrong.

Me: Why not? I thought we’d established that you can’t predict a truly random outcome.

Omega: I don’t need to. I can just mess with your ’s randomness generator so that it gives out pseudo-random numbers instead.

Me: You’re omnipotent now, too?

Omega: Nope. I’ll just give someone a million dollars to do something silly.

Me: No one would ever…! Oh, wait. Anyway, I’ll be able to detect tampering with randomness, the same way it’s possible with a Mersenne twister….

Omega: And I know exactly how soon you’ll give up. Oh, and don’t waste page space suggesting secondary and tertiary levels of ensuring randomness. If, to guide your behavior, you’re using the table of random numbers that I already have, then I already know what you’d do.

Me: Is there any way at all of outsmarting you and getting 1001 K?

Omega: Not one you can find.

Me: Okay then… let me consult smarter people.

This conversation is obviously not going my way. Any suggestions for Act III?

• 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.

• You’re essentially engaging in arbitrage, taking advantage of the difference in the probabilities assigned to both boxes being full by different people. Which is one reason rational people never assign 0 probability to anything.

You could just as well go to some one-boxers (who “believe P(both full) = 0”) and offer them a \$1 bet 10000000:1 in your favor that both boxes will be full; then offer the two-boxers whatever bet they will take “that only one box is full” that will give you more than \$1 profit if you win. Thus, either way, you make a profit, and you can make however much you like just by increasing the stakes.

This still doesn’t actually solve newcomb’s problem, though. I’d call it more of a cautionary tale against being absolutely certain.

(Incidentally, since you’re going into this “fully intending” to take both boxes, I’d expect both one boxers and two boxers to agree on the extremely low probability Omega is going to have filled both boxes.)

• Yes, nshepperd, my assumption is that P << 0.5, something in the 0.0001 to 0.01 range.

Besides, arbitrage would still be possible if some people estimated P=0.01 and others P=0.0001, only the solution would be messier than what I’d ever want to do casually. Besides, if I were unconstrained by the bets I could make (I’d tried to work with a cap before), that would make making profits even easier.

I wasn’t exactly trying to solve the problem, only to find a “naively rational” workaround (using the same naive rationality that leads prisoners to rat each other out in PD).

When you’re saying that this doesn’t solve Newcomb’s problem, what do you expect the solution to actually entail?

• Yes, arbitrage is possible pretty much whenever people’s probabilities disagree to any significant degree. Setting P = 0 just lets you take it to absurd levels (eg. put up no stake at all, and it’s still a “fair bet”).

When you’re saying that this doesn’t solve Newcomb’s problem, what do you expect the solution to actually entail?

Maximizing the money found upon opening the box(es) you have selected.

If you like, replace the money with cures for cancer with differing probabilities of working, or machines with differing probabilities of being a halting oracle, or something else you can’t get by exploiting other humans.

• Which is one reason rational people never assign 0 probability to anything.

I don’t know, I feel pretty confident assigning P(A&!A)=0 :P

• Do you assign 0 probability to the hypothesis that there exists something which you believe to be mathematically true which is not?

• No, P(I’m wrong about something mathematical) is 1-epsilon. P(I’m wrong about this mathematical thing) is often low- like 2%, and sometimes actually 0, like when discussing the intersection of a set and its complement. It’s defined to be the empty set- there’s no way that it can fail to be the empty set. I may not have complete confidence in the rest of set theory, and I may not expect that the complement of a set (or the set itself) is always well-defined, but when I limit myself to probability measures over reasonable spaces then I’m content.

• So, for some particular aspects of math, you have certainty 1-epsilon, where epsilon is exactly zero?

What you are really doing is making the claim “Given that what I know about mathematics is correct, then the intersection of a set and its complement is the empty set.”

• I was interpreting “something” as “at least one thing.” Almost surely my understanding of mathematics as a whole is incorrect somewhere, but there are a handful of mathematical statements that I believe with complete metaphysical certitude.

What you are really doing is making the claim “Given that what I know about mathematics is correct, then the intersection of a set and its complement is the empty set.”

“Correct” is an unclear word, here. Suppose I start off with a handful of axioms. What is the probability that one of the axioms is true /​ correct? In the context of that system, 1, since it’s the starting point. Now, the axioms might not be useful or relevant to reality, and the axioms may conflict and thus the system isn’t internally consistent (i.e. statements having probability 0 and 1 simultaneously). And so the geometer who is only 1-epsilon sure that Euclid’s axioms describe the real world will be able to update gracefully when presented with evidence that real space is curved, even though they retain the same confidence in their Euclidean proofs (as they apply to abstract concepts).

Basically, I only agree with this post when it comes to statements about which uncertainty is reasonable. If you require 1-epsilon certainty for anything, even P(A|A), then you break the math of probability.

• The map is not the territory. “A&!A” would mean some fact about the world being both true and false, rather than anyone’s beliefs about that fact.

• Assigning zero or nonzero probability to that assertion is having a belief about it.

• Yes, the probability is a belief, but your previous question was about something more like P(!A&P(A)=1), that is to say, an absolute belief being inconsistent with the facts. Vaniver’s assertion was about the facts themselves being inconsistent with the facts, which would have a rather alarming lack of implications.

• “Pretty confident” is about as close to “actually 0″ as the moon is (which I don’t care to quantify :P).

• “Pretty confident” is about as close to “actually 0″ as the moon is (which I don’t care to quantify :P).

“Pretty confident” was also a rhetorical understatement. :P

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

How is there anybody in this group? Considering that all of them have \$1,000,000, what convinced them to one-box in the first place such that they later changed their minds about it and regretted the decision? (Like, I guess a one-boxer could say afterwards “I bet that guy wasn’t really omniscient, I should have taken the other box too, then I’d have gotten \$1,001,000 instead”, but why wouldn’t a person who thinks that way two-box to begin with?)

• True.

I only took that case into account for completeness, to cover my bases against the criticism that “not all one-boxers would be happy with their decisions.”

Naively, when you have a choice between 1000000.01 and 1000000.02, it’s very easy to argue that the latter is the better option. To argue for the former, you would probably cite the insignificance of that cent next to the rest of 1000000.01: that eps doesn’t matter, or that an extra penny in your pocket is inconvenient, or that you already have 1000000.01, so why do you need another 0.01?

• 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.

• 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.

dankane, Eliezer answered your question in this comment, and maybe somewhere else, too, that I don’t yet know of.

• If he wasn’t really talking about infinities, how would you parse this comment (the living forever part):

“There is no finite amount of life lived N where I would prefer a 80.0001% probability of living N years to an 0.0001% chance of living a googolplex years and an 80% chance of living forever.”

At very least this should imply that for every N there is an f(N) so that he would rather have a 50% chance of living f(N) years and a 50% chance of dying instantly than having a 100% chance of living for N years. We could then consider the game where if he is going to live for N years he is repeatedly offered the chance to instead live f(N) years with 50% probability and 0 years with 50% probability. Taking the bet n+1 times clearly does better than taking it n times, but the strategy “take the bet until you lose” guarantees him a very short life expectancy.

If your utility function is unbounded you can run into paradoxes like this.

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

Omega #1: (and I think this one is the most plausible) You are given a multiple choice personality test (not knowing what’s about to happen). You are then told that you are in a Newcomb situation and that Omega’s prediction is based on your test answers (maybe they’ll even show you Omega’s code after the test is over). Here I’ll two-box. If I am punished I am not being 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 first case directly contradicts the specifications of the problem, since the idea then becomes to imagine you were the sort of person who would one-box and answer like that, then two box. This might not work for everyone, but a sufficiently clever agent should manage it.

If you are imagining a personality test undertaken in secret, or before you knew you were facing Newcomb’s problem, and stating you would two-box, then it seems like you one-box when it is absolutely certain that omega is right, but two-box if you can think of some way (however unlikely) that he might be wrong.

If you don’t see the problem with this then I suggest you read some of the sequence posts about absolute certainty.

• In the first case, I image the test undertaken in secret. Or more realistically Omega measures these personality traits from listening to my conversations, or reading things I post online.

I don’t decide based on whether there is a possibility that Omega is wrong. #2 can certainly be wrong (for example if I decide based on coin flip) and even #3 can probably mess up. My point is that in case #1 the argument from the post no longer works. If I two-boxed and didn’t get \$1M, I might envy another person for their personality traits (which correlate with one-boxing), but not their decision to one-box.

I think what I am trying to do is split Omega’s decision procedure into cases where either:

• His prediction is clearly caused by my decision (so I should one-box)

• His prediction is not caused by my decision (and so I can two-box without regretting my choice)

(#2 is a special case where I try to be clever.)

• Okay, I misunderstood you.

Even now, I think I would still one-box in case#1. For one thing, it is clearly in my interests, thinking about the problem in advance, to resolve to do so, since the personality test will reveal this fact and I will get the million.

Would you agree with me that far? If so, how do you handle the problem that you seem to be making different decisions at different times, without receiving any new information in between.

• Do you really think that merely deciding to one-box in such a situation would change your personality in a way that gets picked up by the test? If it does, do you want to modify your personality in a measurable way just so that you can win if you happen to run into a Newcomb problem?

Suppose for example it had been determined empirically that whether or not one was religious correlated well with the number of boxes you took. This could then be one of the things that the personality test measures. Are you saying that a precommitment would change your religious beliefs, or that you would change them in addition to deciding to one-box (in which case, why are you changing the latter at all)?

The point in case 1 is that they are not making a direct measurement of your decision. They are merely measuring external factors so that for 99% of people these factors agree with their decision (I think that this is implausible, but not significantly more implausible than the existence of Omega in the first place). It seems to me very unlikely that just changing your mind on whether you should one-box would also automatically change these other factors. And if it does, do you necessarily want to be messing around with your personality just to win this game that will almost certainly never come up?

• If merely deciding to one-box is not picked up by the test, and does not offer even a slight increase in the probability that the money is there (even 51% as opposed to 50% would be enough) then the test is not very good, in which case I would two-box. However, this seems to contradict the stated fact the Omega is in fact a very good predictor of decisions.

As a general principle, I am most definitely interested in modifying my personality to increase the number of situations in which I win. If I wasn’t, I probably wouldn’t be on LW. The religion example is a strawman, as it seems clear that applying the modification “believe in God” will cause me to do worse in many other much more common situations, whereas “one-box in Newcomb-type dilemma’s” doesn’t seem likely to have many side effects.

If Omega really is just measuring external factor’s, then how do you know he won’t pick up on my decision to always one-box. The decision was not made in a vacuum, it was caused by my personality, my style of thinking and my level of intelligence, all of which are things hat any reasonably competent predictor should pick up on.

As long as the test is reasonably good, I will still my million with a higher probability, and that’s all that really matters to me.

• How about this version of Omega (and this is one that I think could actually be implemented to be 90% accurate). First off, box A is painted with pictures of snakes and box B with pictures of bananas. Omega’s prediction procedure is (and you are told this by the people running the experiment) that if you are a human he predicts that you two-box and if you are a chimpanzee, he predicts that you one-box.

I don’t think that 10% of people would give up \$1000 to prove Omega wrong, and if you think so, why not make it \$10^6 and \$10^9 instead of \$10^3 and \$10^6.

I feel like this version satisfies the assumptions of the problem and makes it clear that you should two-box in this situation. Therefore any claims that one-boxing is the correct solution need to at least be qualified by extra assumptions about how Omega operates.

• In this version Omega may be predicting decision’s in general with some accuracy, but it does not seem like he is predicting mine.

So it appears there are cases where I two-box. I think in general my specification of a Newcomb-type problem, has two requirements:

An outside observer who observed me to two-box would predict with high-probability that the money is not there. An outside observer who observed me to one-box would predict with high-probability that the money is there.

The above version of the problem clearly does not meet the second requirement.

If this is what you meant by your statement that the problem is ambiguous, then I agree. This is one of the reasons I favour a formulation involving a brain-scanner rather than a nebulous godlike entity, since it seems more useful to focus on the particularly paradoxical cases rather than the easy ones.

• I don’t think that you change of just that decision would be picked up on a personality test. Your changing that decision is unlikely to change how you answer questions not directly relating to Newcomb’s problem. The test would pick up your style of thinking that lead you to this decision, but making the decision differently would not change your style of thinking. Perhaps an example that illustrates my point even better:

Omega #1.1: Bases his prediction on a genetic test.

Now I agree that it is unlikely that this will get 99% accuracy, but I think it could plausibly obtain, say, 60% accuracy, which shouldn’t really change the issue at hand. Remember that Omega does not need to measure things that cause you to decide one way or another, he just needs to measure things that have a positive correlation with it.

As for modifying your personality… Should I really believe that you believe that arguments that you are making here, or are you just worried that you are going to be in this situation and that Omega will base his prediction on your posts?

• Good point with the genetic test argument, in that situation I probably would two-box. The same might apply to any sufficiently poor personality test, or to a version of Omega that bases his decision of the posts I make on Less Wrong (although I think if my sole reason for being here was signalling my willingness to make certain choices in certain dilemma’s I could probably find better ways to do it).

I usually imagine Omega does better than that, and that his methods are at least as sophisticated as figuring out how I make decisions, then applying that algorithm to the problem at hand (the source of this assumption is that the first time I saw the problem Omega was a supercomputer that scanned people’s brains).

As for the personality modification thing, I really don’t see what you find so implausible about the idea that I’m not attached to my flaws, and would eliminate them if I had the chance.

• I agree that the standard interpretation of Omega generally involves brain scans. But there is still a difference between running a simulation (Omega #2), or checking for relevant correlating personality traits. The later I would claim is at least somewhat analogous to genetic testing, though admittedly the case is somewhat murkier. I guess perhaps the Omega that is most in the spirit of the question is where he does a brain scan and searches for your cached answer of “this is what I do in Newcomb problems”.

As for personality modification, I don’t see why changing my stored values for how to behave in Newcomb situations would change how I behave in non-Newcomb situations. I also don’t see why these changes would necessarily be an improvement.

• “I don’t see why changing my stored values for how to behave in Newcomb situations would change how I behave in non-Newcomb situations.”

It wouldn’t, that’s the point. But it would improve your performance in Newcomb situations, so there’s no downside (for an example of a newcomb type paradox which could happen in the real world, see Parfit’s hitch-hiker, given that I am not a perfect liar I would not consider it too unlikely that I will face a situation of that general type (if not that exact situation) at some point in my life).

• My point was that if it didn’t change your behavior in non-Newcomb situations, no reasonable version of Omega #1 (or really any Omega that does not use either brain scans or lie detection could tell the difference).

As for changing my actions in the case of Parfit’s hitch-hiker, say that the chances of actually running into this situation (with someone who can actually lie detect and in a situation with no third alternatives, and where my internal sense of fairness wouldn’t just cause me to give him the \$100 anyway) is say 10^-9. This means that changing my behavior would save me an expected say 3 seconds of life. So if you have a way that I can actually precommit myself that takes less than 3 seconds to do, I’m all ears.

• It wouldn’t have to be that exact situation.

In fact, it is applicable in any situation where you need to make a promise to someone who has a reasonable chance of spotting if you lie (I don’t know about you but I often get caught out when I lie), and while you prefer following through on the promise to not making it, you also prefer going back on the promise to following through on it, (technically they need to have a good enough chance of spotting you, with “good enough” determined by your relative preferences).

That’s quite a generic situation, and I would estimate at least 10% probability that you encounter it at some point, although the stakes will hopefully be lower than your life.

• Perhaps. Though I believe that in the vast majority of these cases my internal (and perhaps irrational) sense of fairness would cause me to keep my word anyway.

• “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?

• It looks like you just restated the “paradox”—using one argument, it is “obvious” to pick B and using another argument, it is “obvious” to pick both.

Also, in general, do try to avoid saying something is “obvious”. It usually throws a lot of complexity and potential faults into a black box and worsens your chances of uncovering those faults by intimidating people.

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

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

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

Edit I’ve given the Math in a comment below to show how to calculate when to make either 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…

• So working the math on that

Let P(BS) = probability of a Black Swan being involved

This makes the average payout work out to:

1-Box = \$1,000,000

2-Box = \$1,001,000 (1 - P(BS)) + \$1,000 P(BS)

Now it seems to be that the average 2-boxer is assuming that P(BS) = 0, which would make the 2-Box solution always == \$1,001,000 which would, of course, always beat the 1-box solution.

and maybe in this toy-problem, they’re right to assume P(BS) = 0 But IRL that’s almost never the case—after all, 0 is not a probability yes?

So assume that P(BS) is non-zero. t what point would it be worth it to choose the 1-Box solution and what point the 2-Box solution? Lets run the math:

1,000,000 = 1,001,000(1-x) + 1000x = 1001000 − 1001000x + 1000x = 1001000 - (1002000x)

=> 1000000 − 1001000 = −1002000x

=> x = −1000/​-100200

=> x = 0.000998004

So, the estimated probability of Black Swan existing only has to be greater than 0.0998% for the 1-Box solution to have a greater expected payout and therefore the 1-Box option is the more rational::Bayesian choice

OTOH, if you can guarantee that P(BS) is less than 0.0998%, then the rational choice is to 2-Box.

• I’m not sure what you are implying with this link—can you please expand? Are you saying that I’m choosing a least convenient possible world (and if so, how and what) or that 2-boxers are doing so?

• Sorry, your comment was confusing and I didn’t properly concentrate on what you meant, so giving the LCPW link was a mistake, it doesn’t seem to apply.

• No problem. I’ve expanded with the math explaining what I mean, hopefully that makes it less confusing what I was aiming at.

• You are finding technical flaws that are not essential to the intended sense of the thought experiment. Instead of making it uninteresting because of the potential flaws, make the thought experiment stronger by considering the case where these flaws are fixed.

• 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.

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

• You said:

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

but you also said:

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

I can envision several possibilities:

• Perhaps you changed your mind and presently disagree with one of the above two statements.

• Perhaps you didn’t mean a causal AI in the second quote. In that case I have no idea what you meant.

• Perhaps Newcomb’s problem is the wrong example, and there’s some other example motivating TDT that a self-modifying causal agent would deal with incorrectly.

• Perhaps you have a model of causal decision theory that makes self-modification impossible in principle. That would make your first statement above true, in a useless sort of way, so I hope you didn’t mean that.

Would you like to clarify?

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

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

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

• Okay, thanks for confirming that Newcomb’s problem is a relevant motivating example here.

“I don’t know how to solve this problem, nor do I understand the exact structure of the calculation my computer program will perform in the course of solving this problem, nor can I state a mathematically precise meta-question, but I’m going to rely on the AI solving it for me ’cause it’s supposed to be super-smart,”

I’m not saying that. I’m saying that self-modification solves the problem, assuming the CDT agent moves first, and that it seems simple enough that we can check that a not-very-smart AI solves it correctly on toy examples. If I get around to attempting that, I’ll post to LessWrong.

Assuming the CDT agent moves first seems reasonable. I have no clue whether or when Omega is going to show up, so I feel no need to second-guess the AI about that schedule.

(Quoting out of order)

This is not only ugly...

As you know, we can define a causal decision theory agent in one line of math. I don’t know a way to do that for TDT. Do you? If TDT could be concisely described, I’d agree that it’s the less ugly alternative.

but also has worse implications for e.g. meeting an alien AI who wants to cooperate with you, or worse, an alien AI that is trying to blackmail you.

I’m failing to suspend disbelief here. Do you have motivating examples for TDT that seem likely to happen before Kurzweil’s schedule for the Singularity causes us to either win or lose the game?

• As you know, we can define a causal decision theory agent in one line of math.

If you appreciate simplicity/​elegance, I suggest looking into UDT. UDT says that when you’re making a choice, you’re deciding the output of a particular computation, and the consequences of any given choice are just the logical consequences of that computation having that output.

CDT in contrast doesn’t answer the question “what am I actually deciding when I make a decision?” nor does it answer “what are the consequences of any particular choice?” even in principle. CDT can only be described in one line of math because the answer to the latter question has to be provided to it via an external parameter.

• Thanks, I’ll have a look at UDT.

CDT can only be described in one line of math because the answer to the latter question has to be provided to it via an external parameter.

I certainly agree there.

• If TDT could be concisely described, I’d agree that it’s the less ugly alternative.

Maybe this one: “Argmax[A in Actions] in SumO in Outcomes*P(this computation yields A []-> O|rest of universe)”

From this post.

• but also has worse implications for e.g. meeting an alien AI who wants to cooperate with you, or worse, an alien AI that is trying to blackmail you.

I’m failing to suspend disbelief here. Do you have motivating examples for TDT that seem likely to happen before Kurzweil’s schedule for the Singularity causes us to either win or lose the game?

I’m reasonably sure Eliezer meant implications for the would-be friendly AI meeting alien AIs. That could happen at any time in the remaining life span of the universe.

• Causal decision theorists don’t self-modify to timeless decision theorists.

Why not? A causal decision theorist can have an accurate abstract understanding of both TDT and CDT and can calculate the expected utility of applying either. If TDT produces a better expected outcome in general then it seems like self modifying to become a TDT agent is the correct decision to make. Is there some restriction or injunction assumed to be in place with respect to decision algorithm implementation?

Thinking about it for a a few minutes: It would seem that the CDT agent will reliably update away from CDT but that the new algorithm will be neither CDT or TDT (and not UDT either). It will be able to cooperate with agents when there has been some sort causal entanglement between the modified source code and the other agent but not able to cooperate with complete strangers. The resultant decision algorithm is enough of an attractor that it deserves a name of its own. Does it have one?

• Doesn’t have a name as far as I know. But I’m not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch? Maybe, now that I think about it, but even so, does it matter?

A causal decision theorist can have an accurate abstract understanding of both TDT and CDT and can calculate the expected utility of applying either.

But it will calculate that expected value using CDT!expectation, meaning that it won’t see how self-modifying to be a timeless decision theorist could possibly affect what’s already in the box, etcetera.

• Doesn’t have a name as far as I know. But I’m not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch?

Yes, because there are lemmas you can prove about (some) decision theory problems which imply that CDT and UDT give the same output. For example, CDT works if there is exists a total ordering over inputs given to the strategy, common to all execution histories, such that the world program invokes the strategy only with increasing, non-repeating inputs on that ordering. There are (relatively) easy algorithms for these cases. CDT in general is then a matter of applying a theorem when one of its preconditions doesn’t hold, which is one of the most common math mistakes ever.

• Is that really so bad, if it takes the state of the world at the point before it self-modifies as an unchangeable given, and self-modifies to a decision theory that only considers states from that point on as changeable by its decision theory? For one thing, doesn’t that avoid Roko’s basilisk?

• Is that really so bad, if it takes the state of the world at the point before it self-modifies as an unchangeable given, and self-modifies to a decision theory that only considers states from that point on as changeable by its decision theory?

If you do that, you’d be vulnerable to extortion from any other AIs that happen to be created earlier in time and can prove their source code.

• I’m inclined to think that in most scenarios the first AGI wins anyway. And leaving solving decision theory to the AGI could mean you get to build it earlier.

• I’m inclined to think that in most scenarios the first AGI wins anyway.

I was thinking of meeting alien AIs, post-Singularity.

And leaving solving decision theory to the AGI could mean you get to build it earlier.

Huh? I thought we were supposed to be the good guys here? ;-)

But seriously, “sacrifice safety for speed” is the “defect” option in the game of “let’s build AGI”. I’m not sure how to get the C/​C outcome (or rather C/​C/​C/​...), but it seems too early to start talking about defecting already.

Besides, CDT is not well defined enough that you can implement it even if you wanted to. I think if you were forced to implement a “good enough” decision theory and hope for the best, you’d pick UDT at this point. (UDT is also missing a big chunk from its specifications, namely the “math intuition module” but I think that problem has to be solved anyway. It’s hard to see how an AGI can get very far without being able to deal with logical/​mathematical uncertainty.)

• I was thinking of meeting alien AIs, post-Singularity.

What pre-singularity actions are you worried about them taking?

Huh? I thought we were supposed to be the good guys here? ;-)

What I was thinking was that a CDT-seeded AI might actually be safer precisely because it won’t try to change pre-Singularity events, and if it’s first the new decision theory will be in place in time for any post-Singularity events.

Besides, CDT is not well defined enough that you can implement it even if you wanted to.

That’s surprising to me—what should I read in order to understand this point better? EDIT: strike that, you answer that above.

• What pre-singularity actions are you worried about them taking?

They could modify themselves so that if they ever encounter a CDT-descended AI they’ll start a war (even if it means mutual destruction) unless the CDT-descended AI gives them 99% of its resources.

• They could modify themselves so that if they ever encounter a CDT-descended AI they’ll start a war (even if it means mutual destruction) unless the CDT-descended AI gives them 99% of its resources.

They could also modify themselves to make the analogous threat if they encounter a UDT-descended AI, or a descendant of an AI designed by TIm Freeman, or a descendant of an AI designed by Wei Dai, or a descendant of an AI designed using ideas mentioned on LessWrong. I would hope that any of those AI’s would hand over 99% of their resources if the extortionist could prove its source code and prove that war would be worse. I assume you’re saying that CDT is special in this regard. How is it special?

(Thanks for the pointer to the James Joyce book, I’ll have a look at it.)

• I assume you’re saying that CDT is special in this regard. How is it special?

If the alien AI computes the expected utility of “provably modify myself to start a war against CDT-AI unless it gives me 99% of its resources”, it’s certain to get a high value, whereas if it computes the expected utility of “provably modify myself to start a war against UDT-AI unless it gives me 99% of its resources” it might possibly get a low value (not sure because UDT isn’t fully specified), because the UDT-AI, when choosing what to do when faced with this kind of threat, would take into account the logical correlation between its decision and the alien AI’s prediction of its decision.

• ...if it computes the expected utility of “provably modify myself to start a war against UDT-AI unless it gives me 99% of its resources” it might possibly get a low value (not sure because UDT isn’t fully specified), because the UDT-AI, when choosing what to do when faced with this kind of threat, would take into account the logical correlation between its decision and the alien AI’s prediction of its decision.

Well, that’s plausible. I’ll have to work through some UDT examples to understand fully.

What model do you have of how entity X can prove to entity Y that X is running specific source code?

The proof that I can imagine is entity Y gives some secure hardware Z to X, and then X allows Z to observe the process of X self-modifying to run the specified source code, and then X gives the secure hardware back to Y. Both X and Y can observe the creation of Z, so Y can know that it’s secure and X can know that it’s a passive observer rather than a bomb or something.

This model breaks the scenario, since a CDT playing the role of Y could self-modify any time before it hands over Z and play the game competently.

Now, if there’s some way for X to create proofs of X’s source code that will be convincing to Y without giving advance notice to Y, I can imagine a problem for Y here. Does anyone know how to do that?

(I acknowledge that if nobody knows how to do that, that means we don’t know how to do that, not that it can’t be done.)

Hmm, this explains my aversion to knowing the details of what other people are thinking. It can put me at a disadvantage in negotiations unless I am able to lie convincingly and say I do not know.

• I think I″ll stop here for now, because you already seem intrigued enough to want to learn about UDT in detail. I’m guessing that once you do, you won’t be so motivated to think up reasons why CDT isn’t really so bad. :) Let me know if that turns out not to be the case though.

• What model do you have of how entity X can prove to entity Y that X is running specific source code?

On second thought, I should answer this question because it’s of independent interest. If Y is sufficiently powerful, it may be able to deduce the laws of physics and the initial conditions of the universe, and then obtain X’s source code by simulating the universe up to when X is created. Note that Y may do this not because it wants to know X’s source code in some anthropomorphic sense, but simply due to how its decision-making algorithm works.

• If Y is sufficiently powerful, it may be able to deduce the laws of physics and the initial conditions of the universe, and then obtain X’s source code by simulating the universe up to when X is created.

Unless there have been some specific assumptions made about the universe that will not work. Simulating the entire universe does not tell Y which part of the universe it inhabits. It will give Y a set of possible parts of the universe which match Y’s observations. While the simulation strategy will allow the best possible prediction about what X’s source code is given what Y already knows it does not give evidence to Y that it didn’t already have.

• You’re right, the model assumes that we live in a universe such that superintelligent AIs would “naturally” have enough evidence to infer the source code of other AIs. (That seems quite plausible, although by no means certain, to me.) Also, since this is a thread about the relative merits of CDT, I should point out that there are some games in which CDT seems to win relative to TDT or UDT, which is a puzzle that is still open.

• Also, since this is a thread about the relative merits of CDT, I should point out that there are some games in which CDT seems to win relative to TDT or UDT, which is a puzzle that is still open.

It’s an interesting problem, but my impression when reading was somewhat similar to that of Eliezer in the replies. At the core it is the question of “How do you deal with constructs made by other agents?” I don’t think TDT has any particular weakness there.

• If Y is sufficiently powerful, it may be able to deduce the laws of physics and the initial conditions of the universe, and then obtain X’s source code by simulating the universe up to when X is created.

Quantum mechanics seems to be pretty clear that true random number generators are available, and probably happen naturally. I don’t understand why you consider that scenario probable enough to be worth talking about.

• It’s hard to see how an AGI can get very far without being able to deal with logical/​mathematical uncertainty.

Do you have an intuition as to how it would do this without contradicting itself? I tried to ask a similar question but got it wrong in the first draft and afaict did not receive an answer to the relevant part.

I just want to know if my own intuition fails in the obvious way.

• Besides, CDT is not well defined enough that you can implement it even if you wanted to. I think if you were forced to implement a “good enough” decision theory and hope for the best, you’d pick UDT at this point.

Really? That’s surprising. My assumption had been that CDT would be much simpler to implement—but just give undesirable outcomes in whole classes of circumstance.

• CDT uses a “causal probability function” to evaluate the expected utilities of various choices, where this causal probability function is different from the epistemic probability function you use to update beliefs. (In EDT they are one and the same.) There is no agreement amongst CDT theorists how to formulate this function, and I’m not aware of any specific proposal that can be straightforwardly implemented. For more details see James Joyce’s The foundations of causal decision theory.

• There is no agreement amongst CDT theorists how to formulate this function, and I’m not aware of any specific proposal that can be straightforwardly implemented.

I understand AIXI reasonably well and had assumed it was a specific implementation of CDT, perhaps with some tweaks so the reward values are generated internally instead of being observed in the environment. Perhaps AIXI isn’t close to an implementation of CDT, perhaps it’s perceived as not specific or straightforward enough, or perhaps it’s not counted as an implementation. Why isn’t AIXI a counterexample?

• Why isn’t AIXI a counterexample?

You may be right that AIXI can be thought of as an instance of CDT. Hutter himself cites “sequential decision theory” from a 1957 paper which certainly predates CDT, but CDT is general enough that SDT could probably fit into its formalism. (Like EDT can be considered an instance of CDT with the causal probability function set to be the same as the epistemic probability function.) I guess I hadn’t considered AIXI as a serious candidate due to its other major problems.

• http://​​www.alife.co.uk/​​essays/​​on_aixi/​​

Four problems are listed there.

The first one is the claim that AIXI wouldn’t have a proper understanding of its body because its thoughts are defined mathematically. This is just wrong, IMO; my refutation, for a machine that’s similar enough to AIXI for this issue to work the same, is here. Nobody has engaged me in serious conversation about that, so I don’t know how well it will stand up. (If I’m right on this, then I’ve seen Eliezer, Tim Tyler, and you make the same error. What other false consensuses do we have?)

The second one is fixed if we do the tweak I mentioned in the grandparent of this comment.

If you take the fix described above for the second one, what’s left of the third one is the claim that instantaneous human (or AI) experience is too nuanced to fit in a single cell of a Turing machine. According to the original paper, page 8, the symbols on the reward tape are drawn from an alphabet R of arbitrary but fixed size. All you need is a very large alphabet and this one goes away.

I agree with the facts asserted in Tyler’s fourth problem, but I do not agree that it is a problem. He’s saying that Kolmogorov complexity is ill-defined because the programming language used is undefined. I agree that rational agents might disagree on priors because they’re using different programming languages to represent their explanations. In general, a problem may have multiple solutions. Practical solutions to the problems we’re faced with will require making indefensible arbitrary choices of one potential solution over another. Picking the programming language for priors is going to be one of those choices.

• The first one is the claim that AIXI wouldn’t have a proper understanding of its body because its thoughts are defined mathematically. This is just wrong, IMO; my refutation, for a machine that’s similar enough to AIXI for this issue to work the same, is here.

I don’t see how your refutation applies to AIXI. Let me just try to explain in detail why I think AIXI will not properly protect its body. Consider an AIXI that arises in a simple universe, i.e., one computed by a short program P. AIXI has a probability distribution not over universes, but instead over environments where an environment is a TM whose output tape is AIXI’s input tape and whose input tape is AIXI’s output tape. What’s the simplest environment that fits AIXI’s past inputs/​outputs? Presumably it’s E = P plus some additional pieces of code that injects E’s inputs into where AIXI’s physical output ports are located in the universe (that is, overrides the universe’s natural evolution using E’s inputs), and extracts E’s outputs from where AIXI’s physical input ports are located.

What happens when AIXI considers an action that destroys its physical body in the universe computed by P? As long as the input/​output ports are not also destroyed, AIXI would expect that the environment E (with its “supernatural” injection/​extraction code) will continue to receive its outputs and provide it with inputs.

Does that make sense?

• (Responding out of order)

Does that make sense?

Yes, but it makes some unreasonable assumptions.

Consider an AIXI that arises in a simple universe, i.e., one computed by a short program P.

An implementation of AIXI would be fairly complex. If P is too simple, then AIXI could not really have a body in the universe, so it would be correct in guessing that some irregularity in the laws of physics was causing its behaviors to be spliced into the behavior of the world.

However, if AIXI has observed enough of the inner workings of other similar machines, or enough of the laws of physics in general, or enough of its own inner workings, the simplest model will be that AIXI’s outputs really do emerge from the laws of physics in the real universe, since we are assuming that that is indeed the case and that Kolmogorov induction eventually works. At that point, imagining that AIXI’s behaviors are a consequence of a bunch of exceptions to the laws of physics is just extra complexity and won’t be part of the simplest hypothesis. It will be part of some less likely hypotheses, and the AI would have to take that risk into account when deciding whether to self-improve.

• Tim, I think you’re probably not getting my point about the distinction between our concept of a computable universe, and AIXI’s formal concept of a computable environment. AIXI requires that the environment be a TM whose inputs match AIXI’s past outputs and whose outputs match AIXI’s past inputs. A candidate environment must have the additional code to inject/​extract those inputs/​outputs and place them on the input/​output tapes, or AIXI will exclude it from its expected utility calculations.

• The candidate environment must have the additional code to inject/​extract those inputs/​outputs and place them on the input/​output tapes, or AIXI will exclude it from its expected utility calculations.

I agree that the candidate environment will need to have code to handle the inputs. However, if the candidate environment can compute the outputs on its own, without needing to be given the AI’s outputs, the candidate environment does not need code to inject the AI’s outputs into it.

Even if the AI can only partially predict its own behavior based on the behavior of the hardware it observes in the world, it can use that information to more efficiently encode its outputs in the candidate environment, so it can have some understanding of its position in the world even without being able to perfectly predict its own behavior from first principles.

If the AI manages to destroy itself, it will expect its outputs to be disconnected from the world and have no consequences, since anything else would violate its expectations about the laws of physics.

This back-and-forth appears to be useless. I should probably do some Python experiments and we then can change this from a debate to a programming problem, which would be much more pleasant.

• However, if the candidate environment can compute the outputs on its own, without needing to be given the AI’s outputs, the candidate environment does not need code to inject the AI’s outputs into it.

If a candidate environment has no special code to inject AIXI’s outputs, then when AIXI computes expected utilities, it will find that all actions have equal utility in that environment, so that environment will play no role in its decisions.

I should probably do some Python experiments and we then can change this from a debate to a programming problem, which would be much more pleasant.

Ok, but try not to destroy the world while you’re at it. :) Also, please take a closer look at UDT first. Again, I think there’s a strong possibility that you’ll end up thinking “why did I waste my time defending CDT/​AIXI?”

• FYI, generating reward values internally—instead of them being observed in the environment—makes no difference whatsoever to the wirehead problem.

AIXI digging into its brains with its own mining claws is quite plausible. It won’t reason as you suggest—since it has no idea that it is instantiated in the real world. So, its exploratory mining claws may plunge in. Hopefully it will get suitably negatively reinforced for that—though much will depend on which part of its brain it causes damage too. It could find that ripping out its own inhibition circuits is very rewarding.

A larger set of symbols for rewards makes no difference—since the reward signal is a scalar. If you compare with an animal, that has millions of pain sensors that operate in parallel. The animal is onto something there—something to do with a-priori knowledge about the common causes of pain. Having lots of pain sensors has positive aspects—e.g. it saves you experimenting to figure out what hurts.

As for the reference machine issue, I do say: “This problem is also not very serious.”

Not very serious unless you are making claims about your agent being “the most intelligent unbiased agent possible”. Then this kind of thing starts to make a difference...

• A larger set of symbols for rewards makes no difference—since the reward signal is a scalar. If you compare with an animal, that has millions of pain sensors that operate in parallel. The animal is onto something there—something to do with a-priori knowledge about the common causes of pain. Having lots of pain sensors has positive aspects—e.g. it saves you experimenting to figure out what hurts.

You can encode 16 64 bit integers in a 1024 bit integer. The scalar/​parallel distinction is bogus.

(Edit: I original wrote “5 32 bit integers” when I meant “2**5 32 bit integers”. Changed to “16 64 bit integers” because “32 32 bit integers” looked too much like a typo.)

Not very serious unless you are making claims about your agent being “the most intelligent unbiased agent possible”. Then this kind of thing starts to make a difference...

Strawman argument. The only claim made is that it’s the most intelligent up to a constant factor, and a bunch of other conditions are thrown in. When Hutter’s involved, you can bet that some of the constant factors are large compared to the size of the universe.

• You can encode 5 32 bit integers in a 1024 bit integer. The scalar/​parallel distinction is bogus.

Er, not if you are adding the rewards together and maximising the results, you can’t! That is exactly what happens to the rewards used by AIXI.

Not very serious unless you are making claims about your agent being “the most intelligent unbiased agent possible”. Then this kind of thing starts to make a difference...

Strawman argument. The only claim made is that it’s the most intelligent up to a constant factor, and a bunch of other conditions are thrown in.

Actually Hutter says this sort of thing all over the place (I was quoting him above) - and it seems pretty irritating and misleading to me. I’m not saying the claims he makes in the fine print are wrong, but rather that the marketing headlines are misleading.

• You can encode 5 32 bit integers in a 1024 bit integer. The scalar/​parallel distinction is bogus.

Er, not if you are adding the rewards together and maximising the results, you can’t! That is exactly what happens to the rewards used by AIXI.

You’re right there, I’m confusing AIXI with another design I’ve been working with in a similar idiom. For AIXI to work, you have to combine together all the environmental stuff and compute a utility, make the code for doing the combining part of the environment (not the AI), and then use that resulting utility as the input to AIXI.

• For more details see James Joyce’s The foundations of causal decision theory.

Thankyou for the reference, and the explanation.

I am prompted to ask myself a question analogous to the one Eliezer recently asked:

Doesn’t have a name as far as I know. But I’m not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch? Maybe, now that I think about it, but even so, does it matter?

Is it worth my while exploring the details of CDT formalization beyond just the page you linked to? There seems to be some advantage to understanding the details and conventions of how such concepts are described. At the same time revising CDT thinking in too much detail may eliminate some entirely justifiable confusion as to why anyone would think it is a good idea! “Causal Expected Utiluty”? “Causal Tendencies”? What the? I only care about what will get me the best outcome!

• Is it worth my while exploring the details of CDT formalization beyond just the page you linked to?

Probably not. I only learned it by accident myself. I had come up with a proto-UDT that was motivated purely by anthropic reasoning paradoxes (as opposed to Newcomb-type problems like CDT and TDT), and wanted to learn how existing decision theories were formalized so I could do something similar. James Joyce’s book was the most prominent such book available at the time.

ETA: Sorry, I think the above is probably not entirely clear or helpful. It’s a bit hard for me to put myself in your position and try to figure out what may or may not be worthwhile for you. The fact is that Joyce’s book is the decision theory book I read, and quite possibly it influenced me more than I realize, or is more useful for understanding the motivation for or the formulation of UDT than I think. It couldn’t hurt to grab a copy of it and read a few chapters to see how useful it is to you.

• Thanks for the edit/​update. For reference it may be worthwhile to make such additions as a new comment, either as a reply to yourself or the parent. It was only by chance that I spotted the new part!

• I was thinking of meeting alien AIs, post-Singularity.

What pre-singularity actions are you worried about them taking?

Huh? I thought we were supposed to be the good guys here? ;-)

What I was thinking was that a CDT-seeded AI might actually be safer precisely because it won’t try to change pre-Singularity events, and if it’s first the new decision theory will be in place in time for any post-Singularity events.

Besides, CDT is not well defined enough that you can implement it even if you wanted to.

That’s surprising to me—what should I read in order to understand this point better?

• But I’m not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch? Maybe, now that I think about it, but even so, does it matter?

Yes, for reasons of game theory and of practical singularity strategy.

Game theory, because things in Everett branches that are ‘closest’ to us might be the ones it’s most important to be able to interact with, since they’re easier to simulate and their preferences are more likely to have interesting overlap with ours. Knowing very roughly what to expect from our neighbors is useful.

And singularity strategy, because if you can show that architectures like AIXI-tl have some non-negligible chance of converging to whatever an FAI would have converged to, as far as actual policies go, then that is a very important thing to know; especially if a non-uFAI existential risk starts to look imminent (but the game theory in that case is crazy). It is not probable but there’s a hell of a lot of structural uncertainty and Omohundro’s AI drives are still pretty informal. I am still not absolutely sure I know how a self-modifying superintelligence would interpret or reflect on its utility function or terms therein (or how it would reflect on its implicit policy for interpreting or reflecting on utility functions or terms therein). The apparent rigidity of Goedel machines might constitute a disproof in theory (though I’m not sure about that), but when some of the terms are sequences of letters like “makeHumansHappy” or formally manipulable correlated markers of human happiness, then I don’t know how the syntax gets turned into semantics (or fails entirely to get turned into semantics, as they case may well be).

But it will calculate that expected value using CDT!expectation, meaning that it won’t see how self-modifying to be a timeless decision theorist could possibly affect what’s already in the box, etcetera.

This implies that the actually-implemented-CDT agent has a single level of abstraction/​granularity at like the naive realist physical level at which it’s proving things about causal relationships. Like, it can’t/​shouldn’t prove causal relationships at the level of string theory, and yet it’s still confident that its actions are causing things despite that structural uncertainty, and yet despite the symmetry it for some reason cannot possibly see how switching a few transistors or changing its decision policy might affect things via relationships that are ultimately causal but currently unknown for reasons of boundedness and not speculative metaphysics. It’s plausible, but I think letting a universal hypothesis space or maybe even just Goedelian limitations enter the decision calculus at any point is going to make such rigidity unlikely. (This is related to how a non-hypercomputation-driven decision theory in general might reason about the possibility of hypercomputation, or the risk of self-diagonalization, I think.)

• But it will calculate that expected value using CDT!expectation, meaning that it won’t see how self-modifying to be a timeless decision theorist could possibly affect what’s already in the box, etcetera.

The CDT is making a decision about whether to self-modify even before it meets the alien, based on its expectation of meeting the alien. How does CDT!expectation differ from Eliezer!expectation before we meet the alien?

• Doesn’t have a name as far as I know. But I’m not sure it deserves one; would CDT really be a probable output anywhere besides a verbal theory advocated by human philosophers in our own Everett branch? Maybe, now that I think about it, but even so, does it matter?

It is useful to separate in one’s mind the difference between on one hand being able to One Box and cooperate in PD with agents that you know well (shared source code) and on the other hand not firing on Baby Eaters after they have already chosen not to fire on you. This is especially the case when first grappling the subject. (Could you confirm, by the way, that Akon’s decision in that particular paragraph or two is approximately what TDT would suggest?)

The above is particularly relevant because the “have access to each other’s source code” is such a useful intuition pump when grappling with or explaining the solutions to many of the relevant decision problems. It is useful to be able to draw a line on just how far the source code metaphor can take you.

There is also something distasteful about making comparisons to a decision theory that isn’t even implicitly stable under self modification. A CDT agent will change to CDT++ unless there is an additional flaw in the agent beyond the poor decision making strategy. If I create a CDT agent, give it time to think and then give it Newcomb’s problem it will One Box (and also no longer be a CDT agent). It is the errors in the agent that still remain after that time that need TDT or UDT to fix.

But it will calculate that expected value using CDT!expectation, meaning that it won’t see how self-modifying to be a timeless decision theorist could possibly affect what’s already in the box, etcetera.

*nod* This is just the ‘new rules starting now’ option. What the CDT agent does when it wakes up in an empty, boring room and does some introspection.

• Surely the important thing is that it will self-modify to whatever decision theory has the best consequences?

The new algorithm will not exactly be TDT, because it won’t try to change decisions that have already been made the way TDT does. In particular this means that there’s no risk from Roko’s basilisk.

Disclaimer: I’m not very confident of anything I say about decision theory.

• Eliezer says elsewhere that current decision theory doesn’t let us prove a self-modifying AI would choose to keep the goals we program into it. He wants to develop a proof before even starting work on the AI.

• It’s easy to contrive situations where a self-modifying AI would choose not to keep the goals programmed into it, even without precommitment issues. Just contrive the circumstances so it gets paid to change. Unless there’s something wrong with the argument there, TDT etc. won’t be enough to ensure that the goals are kept.

• 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

• 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:

The problem is, such emphatic declarations of confidence in the right answer can just as easily be followed by one-boxing, two-boxing, or declaring the hypotheses self-contradictory. That is, in fact, what makes it a Problem, even if, to any individual, it is not a problem.

• Differing outcomes are a problem by themselve. Either one reasoning is right and the others are wrong, or basic logic is broken (and it would follow all maths are broken). It could also be that some hypothesis absolutely necessary for one reasoning or the other are implicit and untelled.

This is why, even if to me Newcomb is not a problem, it is still critical to find where other’s reasoning are either broken or which assumptions are hidden. Failure to exhibit any error in someone else reasoning would lead to conclude that either my reasoning is broken (and I would have to find why) or that maths are broken. And I take that very seriously.

That’s also why when rejecting someone else reasoning stating we believe another well known reasoning is right (authority argument) is never enough. For the sake of rationality we should also find the error (if any) in the other’s reasoning.

So, what is wrong believing in probabilities ?

• So, what is wrong believing in probabilities ?

To ask that question is already to presuppose the one-boxing answer, and to miss the problem that the problem itself may be problematic. I don’t take simple two-boxing any more seriously than Amanojack does, but the third possibility, of disputing that the problem is well-posed, is worth exploring. On LW, self-professed two-boxers are usually taking that alternative. (Elsewhere, I see two-boxing philosophers actually saying that two-boxing loses, but is still the rational thing to do.)

The problem is best disputed not by simply asserting, as some have, that no such Omega can exist, but by thinking in detail about what it would take for someone to predict the decisions of a decision-maker who knows you’re trying to predict their decisions. What that sort of thinking looks like is this. That paper is about Prisoners Dilemma, but similar investigations could be made of Newcomb, Parfit’s Hitchhiker, etc.

That is what fighting the hypothesis looks like, done right.

• So, what is wrong believing in probabilities ? To ask that question is already to presuppose the one-boxing answer, and to miss the problem that the problem itself may be problematic.

That is going for the third option and dodging to point out exactly why the problem should not be well posed. I can write a program working as the Newcomb’s problem is described if I go for the “unperfect predictor” version where the being is merely right “most of the time”. A way to do it could be to let player run a number of practice (or calibration) games, then at a time chosen by the guesser make that game “real”. The calibration plays would simulate the supernatural player minute observation of the player behavior, what can indeed not easily be done.

I knew of the Robust Coopearation paper, and it’s really very interresting, but getting the source code of the other is also a huge change to the initial problem. At least it excludes perfect oracles from the problem, it is also clear you may be confronted to halting problem (this is why current scheme tournament based on this idea had to make a provision in rules to avoid non halting programs). Stating we can say something usefull on another problem does not implies the initial one had anything wrong.

On the other hand, it is obvious that Dominance Argument is broken in Newcom’s problem (and also in PD) as the logical proof is only correct when we have non correlated variables (non correlation should not be confused with causal independance, causal independance is not enough for Dominance Argument to be correct). In Newcomb’s problem, the perfect correlation is part of the problem statement. How anyone could then apply Dominance Argument is beyond me, probably because it mimics usual deductive logic.

I’m not saying that Newcomb’s problem describe any physically possible event, or not even that it is a good problem, or that the consequences it leads to are agreeable (at first sight it leads to lack of free will), but just that mathematically using (very) simple probabilistic tools you can solve it, without changing anything and that alternative usual solution is based on a mathematical error.

• I can write a program working as the Newcomb’s problem is described if I go for the “unperfect predictor” version where the being is merely right “most of the time”. A way to do it could be to let player run a number of practice (or calibration) games, then at a time chosen by the guesser make that game “real”. The calibration plays would simulate the supernatural player minute observation of the player behavior, what can indeed not easily be done.

Are you claiming to be able to write a program that can win, playing Omega in an iterated Newcomb’s Problem contest, basing its predictions only on the record of the outcomes of past training rounds? How high a prediction accuracy do you expect to be able to get?

• As proposed, the idea is to run the program in “test mode”. To simulate the super-being Omenga we give it the opportunity to decide when game stops being a simulation (predictor calibration) and start being the “real game”. To be fair, this change (or the rules governing it) will be communicated to some external judge before the actual “real play”. But it will not be communicated to player (or obviously it would break any calibration accuracy). A possible rule could be to start the real game when some fixed accuracy is reached (something like prediction is right 99% of the time), or it could also be a fixed number of calibration games.

Writing such predictor is as easy as it can be : just a simple loop waiting for the fixed accuration wich is reached when either one-box or two-box is above expected threshold. Obviously it player is random, that could be quite long (but even with random it should happen sometime). But with a rational player that should be fast enough. I’m not sure that a random player could qualify as rational, anyways.

Doing that Omega can be as accurate as wished.

It still is not a perfect predictor, the player could still outguess Omega and predict at wich move the desired accuracy will be reached, but it’s good enough for me (and the Omega player could add some randomness on his side tu avoid guessers).

I see no reason why the program describe above could not be seen as an acceptable Omega following Newcomb’s problem rules.

Not communicating the actual real game is just here to avoid cheaters and enforce that the actual experiment will be done in the same environment sa the calibration.

I wonder if anyone would seriously choose to two-box any time with the above rules.

• To be fair, this change (or the rules governing it) will be communicated to some external judge before the actual “real play”. But it will not be communicated to player (or obviously it would break any calibration accuracy).

But then, the player never knows when they are faced with Omega, the successful predictor, which is an essential part of Newcomb’s problem.

A possible rule could be to start the real game when some fixed accuracy is reached (something like prediction is right 99% of the time), or it could also be a fixed number of calibration games.

Writing such predictor is as easy as it can be : just a simple loop waiting for the fixed accuration wich is reached when either one-box or two-box is above expected threshold. Obviously it player is random, that could be quite long (but even with random it should happen sometime).

You expect to predict even a random choice with 99% accuracy? Am I misunderstanding something? Rock-scissors-paper programs that try to detect the non-randomness of human choices do succeed against most people, but only a little better than chance, not with 99% accuracy. Against a truly random player they do not succeed at all.

But iterated Newcomb is different from original Newcomb, just as iterated PD is different from plain PD. Now, I don’t see anything wrong with studying related problems, but you yourself said that studying a different but related problem does not touch the original.

• I don’t know if you have seen it, but I have posted an actual program playing Newcomb’s game. As far as I understand what I have done, this is not an Iterated Newcomb’s problem, but a single shot one. You should also notice that the calibration phase does not returns output to the player (well, I added some showing of reached accuracy, but this is not necessary).

If I didn’t overviewed some detail, the predictor accuracy is currently tuned at above 90% but any level of accuracy is reachable.

As I explained yesterday, the key point was to run some “calibration” phase before running the actual game. To make the calibration usefull I have to blur the limit between calibration and actual game or the player won’t behave as in real game while in calibration phase. Hence the program need to run a number of “maybe real” games before playing the true one. For the reason explained above we also cannot say to the user he his playing the real and last game (or he would known if he is playing a calibration game or a real one and the calibration would be useless).

But it is very clear reading source code that if the (human) player was some kind of supernatural being he could defeat the program by choosing two boxes while the prediction is one-box. It just will be a very unlikely event to the desired accuracy level.

I pretend this is a true unmodified Newcomb’s problem, all the calibration process is here only to make actually true the preassertion of the Newcomb’s problem : prediction accuracy of Omega (and verifiably so for the human player : he can read the source code and convince himself or even run the program and understand why prediction will be accurate).

As I know it Necomb’s problem does not impose the way the initial preassertion of accuracy is reached. As programming goes, I’m merely composing two functions, the first one ensuring the entry preassertion of good prediction accuracy is true.

• I see a problem with the proposed method. Your program learns how often, on average, its opponent one-boxes or two-boxes. If I (as Omega) learn that someone is a one-boxer, then I can predict that they will one-box next time, put money in box B, and be proved right. But then, in an iterated game, if the one-boxer learns that I am not predicting his decision in the individual case, but have made a general prediction once and for all and thereafter always filling box B, then he can with impunity take both boxes and prove my prediction wrong.

A true Omega needs to make both P(box B full | take one box) and P(box B empty | take both boxes) high. The proposed scheme ensures that P(box B full | habitual one-boxer) and P(box B empty | habitual two-boxer) are high, which is not quite the same.

Similarly, suppose I convince Eliezer that I’m Omega. He has publicly avowed one-boxing on Newcomb, so I can skip the learning phase, fill box B, and be proved right. But if, for some reason, he suspects that I’m not a superintelligent superbeing with superpowers of prediction, and in a series of games, experiments with two-boxing, I will be exposed as an impostor.

Iterated Newcomb played between programs given access to each other’s source code would be an interesting challenge. I assume Omega doesn’t care about the money, but plays for the gratification of correctly predicting the other player’s choice. The other player is playing for the money.

A simpler, zero-sum game also suggests itself to me. This is more like Rock-Paper-Scissors than Newcomb, but again the point is to play using knowledge of the other person’s code. Each player chooses 0 or 1. Player A wins if the choices are the same, player B wins if they are different.

(This might look as if A is trying to predict B and B is trying to avoid being predicted, but the game is actually symmetric, both players doing both of these things. Swap the labels on B’s choices and B wins on equality and A on inequality.)

In classical game theory, the optimal strategy is to toss a coin, and the expected payoff is zero. The challenge is to do better against real opponents.

• A true Omega needs to make both P(box B full | take one box) and P(box B empty | take both boxes) high. The proposed scheme ensures that P(box B full | habitual one-boxer) and P(box B empty | habitual two-boxer) are high, which is not quite the same.

If I understand correctly the distinction you’re making between habitual one boxer and take one box the first kind would be about the past player history and the other one about the future. If so I guess you are right. I’m indeed using the past to make my prediction, as using the future is beyond my reach.

But I believe you’re missing the point. My program is not an iterated Newcomb’s Problem because Omega does not perform any prediction along the way. It will only perform one prediction. And that will be for the last game and the human won’t be warned. It does not care at all about the reputation of the player, but only on it’s acts in situations where he (the human player) can’t know if he is playing of not.

But another point of view is possible, and that is what comes to mind when you run the program: it is coercing the player to be either a one boxer or a two boxer if he wan’t to play at all. Any two-boxing and the player will have to spend a very long time one-boxing to reach the state when he is again seen as a one boxer. As it is written, the program is likely (to the chosen accuracy level) to make it’s prediction while the player is struggling to be a one boxer.

As a human player what comes through my mind while running my program is ok: I want to get a million dollars, henceforth I have to become a one boxer.

• If my program runs as long as wished accuracy is nor reached it can reach any accuracy. Truly random numbers are also expected to deviate toward extremes sometimes in the long run (if they do not behave like that they are not random). As it is very rare events, against random players the expected accuracy would certainly never be reached in a human life.

Why I claim is the “calibration phase” described above takes place before Newcomb’s problem. When the actual game starts the situation described in Newcomb’s problem is exactly what is reached. THe description of the calibration phase could even be provided to the player to convince him Omega prediction will be accurate. At least it is convincing for me and in such a setting I would certaily believe Omega can predict my behavior. In a way you could the my calibration phase as a way for Omega to wait for the player to be ready to play truly instead of trying to cheat. As trying to cheat will only result in delaying the actual play.

OK. It may be another problem, what I did is merely replacing a perfectly accurate being with an infinitely patient one… but this one is easy to program.

• I posted a possible program doing what I describe in another comment. The trick as expected is that it’s easier to change the human player understanding of the nature of omega to reach the desired predictability. In other words : you just remove human free will (and running my program the player learn very quickly that is in his best interrest), then you play. What is interresting is that the only way compatible with Newcomb’s problem description to remove his free will is to make it a one-boxer. The incentive to make it a two-boxer would be to exhibit a bad predictor and that’s not compatible with Newcomb’s problem.

• A way to do it could be to let player run a number of practice (or calibration) games, then at a time chosen by the guesser make that game “real”. The calibration plays would simulate the supernatural player minute observation of the player behavior, what can indeed not easily be done.

No good, then even CDTers are incentivized to one-box, since once-boxing in the practice rounds causes higher rewards in the real rounds.

• I do not see your reasoning here ? What I’m proposing is not letting know when practising round stops and real round starts. That means indeed that one boxer would get higher rewards in both practice and real round, and that’s why I believe it’s an argument for one boxing.

My proposal for “simulating” Newcomb’s may not be accurate (and it’s certainly not perfect) but you can’t conclude that based on the (projected) outcome of the experiment disagreeing with wath you expect.

• Because depending on the numbers in the setup your modified experiment doesn’t get at the disagreement between one-boxer and two-boxers.

• Here is an actual program (written in python) implementing the described experiment. It has two stages. The first part is just calibration intending to find out if the player is one boxing or two boxing. The second is a straightforward non iterated Newcomb problem. Some randomness is used to avoid the player to exactly know when calibration stops and test begin, but calibration part does not care at all if it will predict the player is a one boxer or a two boxer it is just intended to create an actual predictor behaving as described in Newcomb’s.

``````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
``````
• I’m with you. You have to look at the outcomes, otherwise you end up running into the same logical blinders that make Quantum Mechanics hard to accept.

After reading some of the Quantum Mechanics sequence, I am more willing to believe in Omega’s omniscience. Quantum mechanics allows for multiple timelines leading to the same outcome to interfere and simply never happen, even if they would have been probable in classical mechanics. Perhaps all timelines leading to the outcome where `one-boxing does not yield money` happen to interfere. Even if you take a more literal interpretation of the problem statement, where it is your own mind that determines the box’s content, your mind is made of particles which could conceivably affect the universe’s configuration.

• I have more or less the same point of view and applied it to non iterated prisonner’s dilemma (as Newcomb’s is merely half a Prisonner’s Dilemma as David Lewis suggested in an article, and on this I agree with him, but not on his conclusion).

What is at stakes here (in Newcomb’s or PD) may not be that easy to accept anyway. It’s probability and Bayes against causality. The doom loop in Newcomb’s (reasoning loop leading to loose 1 million, as I see it) is stating that The content of the boxes is already put when you play, henceforth you action won’t change anything. The quantum mechanical reasoning would go the other way: as long as you did’nt observe/​interact with it it is merely a probability. You may even want to go futher than that: imagine that someone else see the content of the box, then see you choosing the predicted set of boxes. He will conclude you have no freewill, or something along theses lines. I understand that people puting freewill as a fact—not merely a belief that could be contradicted by experiment—and so reject unthinkingly the probabilist reasoning.

My comment about PD is in this Sequence (http://​​lesswrong.com/​​lw/​​hl8/​​other_prespective_on_resolving_the_prisoners/​​). I merely applty probability rules. I’m interrested to know if you see any fault in it from a probabilist point of view.

• Do you also choose not to chew gum in Eliezer’s version of Solomon’s Problem?

• …You know that paper goes on to assert that the two problems are meaningfully different, such that it’s rational to both one-box in Newcomb’s Problem and chew gum in Solomon’s Problem, right?

• Newcomb’s Problem is silly. It’s only controversial because it’s dressed up in wooey vagueness.

Alternately: It’s entirely precise and well formed, far more so than just about every real life decision.

• Is there something wrong with my argument above?

• It’s only controversial because it’s dressed up in wooey vagueness

I also happen to think that under-specification of this puzzle adds significantly to the controversy.

What the puzzle doesn’t tell us is the properties of the universe in which it is set. Namely, whether the universe permits future to influence the past, which I’ll refer to as “future peeking”.

(alternatively, whether the universe somehow allows someone within the universe to precisely simulate the future faster than it actually comes—a proposition I don’t believe is ever true in any universe defined mathematically).

This is important because if the future can’t influence the past, then it is known with absolute certainty that taking two boxes won’t possibly change what’s in them (this is, after all, a basic given of the universe). Whether Omega has predicted something before is completely irrelevant now that the boxes are placed.

Alas, we aren’t told what the universe is like. If that is intentionally part of the puzzle then the only way to solve it would be to enumerate all possible universes, assigning each one a probability of being ours based on all the available evidence, and essentially come up with a probability that “future peeking” is impossible in our universe. One would then apply simple arithmetic to calculate the expected winnings.

Unfortunately P(“future peeking allowed”) it’s one of those probabilities that is completely incalculable for any practical purpose. Thus if “no future peeking” isn’t a given, the best answer is “I don’t know if taking two boxes is best because there’s this one probability I can’t actually calculate in practice”.

• whether the universe somehow allows someone within the universe to precisely simulate the future faster than it actually comes—a proposition I don’t believe is ever true in any universe defined mathematically

As near as I can tell, this depends on dubious assumptions about a mathematical universe. You appear to treat time as fundamental, and yet reject the possibility that reality (or the Matrix) simulates a certain outcome happening at a certain time, not before (as we’d expect if reality calculated the output of a time-dependent wavefunction).

In addition, you seem to assume that reality cares about the same aspects of the situation that interest Omega. Otherwise it seems clear that Omega could get an answer sooner by leaving out all the details which don’t affect the human-level outcome.

• Assume no “future peeking” and Omega only correctly predicting people as difficult to predict as you with 99.9% probability. One-boxing still wins.

• While I disagree that one-boxing still wins, I’m most interested in seeing the “no future peeking” and the actual Omega success rate being defined as givens. It’s important that I can rely on the 99.9% value, rather than wondering whether it is perhaps inferred from their past 100 correct predictions (which could, with a non-negligible probability, have been a fluke).

• That does indeed seem like the standard version of Newcomb’s. (Though I don’t understand your last sentence, assuming “non-negligible” does not mean 12 to the power of 100.)

Can you spell out what you mean by “if” in this context? Because a lot of us are explicitly talking about the best algorithm to program into an AI.

• Why is it important to you that the success rate be a frequentialist probability rather than just a bayesian one?

• I’m not sure I understand correctly, but let me phrase the question differently: what sort of confidence do we have in “99.9%” being an accurate value for Omega’s success rate?

From your previous comment I gather the confidence is absolute. This removes one complication while leaving the core of the paradox intact. I’m just pointing out that this isn’t very clear in the original specification of the paradox, and that clearing it up is useful.

To explain why it’s important, let me indeed think of an AI like hairyfigment suggested. Suppose someone says they have let 100 previous AIs flip a fair coin 100 times each and it came out heads every single time, because they have magic powers that make it so. This someone presents me video evidence of this feat.

If faced with this in the real world, an AI coded by me would still bet close to 50% on tails if offered to flip its own fair coin against this person, because I have strong evidence that this someone is a cheat, and their video evidence is fake. Just something I know from a huge amount of background information that was not explicitly part of this scenario.

However, when discussing such scenarios, it is sometimes useful to assume hypothetical scenarios unlike the real world. For example, we could state that this someone has actually performed the feat, and that there is absolutely no doubt about that. That’s impossible in our real world, but it’s useful for the sake of discussing bayesianism. Surely any bayesianist’s AI would expect heads with high probability in this hypothetical universe.

So, are we looking at “Omega in the real world where someone I don’t even know tells me they are really damn good at predicting the future”, or “Omega in some hypothetical world where they are actually known with absolute certainty to be really good at predicting the future”?

• Omega has been correct on each of 100 observed occasions so far—everyone who took both boxes has found box B empty and received only a thousand dollars; everyone who took only box B has found B containing a million dollars.

Seems to me the language of this rules out faked video. And to explain it as a newsletter scam would, I think, require postulating 2^100 civilizations that have contact with Omega but not each other. Note that we already have some reason to believe that a powerful and rational observer could predict our actions early on.

So you tell me what we should expect here.

• I’ve reviewed the language of the original statement and it seems that the puzzle is set in essentially the real world with two major givens, i.e. facts in which you have 100% confidence.

Given #1: Omega was correct on the last 100 occurrences.

There is no leeway left for quantum effects, or for your choice affecting in any way what’s in box B. You cannot make box B full by consciously choosing to one-box. The puzzle says so, after all.

If you read it like this, then I don’t see why you would possibly one-box. Given #2 already implies the solution. 100 successful predictions must have been achieved through a very low probability event, or a trick, e.g by offering the bet only to those people whose answer you can already predict, e.g. by reading their LessWrong posts.

If you don’t read it like this, then we’re back to the “gooey vagueness” problem, and I will once again insist that the puzzle needs to be fully defined before it can be attempted. For example, by removing both givens, and instead specifying exactly what you know about those past 100 occurrences. Were they definitely not done on plants? Was there sampling bias? Am I considering this puzzle as an outside observer, or am I imagining myself being part of that universe—in the latter case I have to put some doubt into everything, as I can be hallucinating. These things matter.

With such clarifications, the puzzle becomes a matter of your confidence in the past statistics vs. your confidence about the laws of physics precluding your choice from actually influencing what’s in box B.

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.

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

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

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

If my brain is wired up in such a way as to choose two-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.

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

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

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

• Welcome to Less Wrong!

As a rationalist, there should only ever be one choice you make.

Why do you think so?

• As a rationalist, there should only ever be one choice you make.

I think so too.

if someone knows your exact situation (trivial in the Newcomb paradox, as the super intelligent agent is causing your situation)

Perhaps we’ve all heard a slightly different wording of the paradox (or more), but I don’t see what causation has to do with it.

• Perhaps we’ve all heard a slightly different wording of the paradox (or more), but I don’t see what causation has to do with it.

He knows what your environmental circumstances are because he put you in them. That is, he obviously knows that you are going to be encountering a Newcomblike problem because he just gave it to you. (ie. No deep technical meaning, just the obvious.)

• Maybe I’m being dense. Omega needs to know more than just that you are going to encounter the problem, even Omega’s scheduler and publicist know that!

Omega knows the exact situation, including how an identical model of you would act/​has acted, because that is stipulated, but it does not follow trivially from Omega’s causing your situation.

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

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

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

In both cases, I should one-box. So I’ll one-box. I don’t really get the 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.

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

This is such an important concept.

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

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

• I liked the quote not because of any notion that Bayes will or should “go out the window,” but because, coming from a devout (can I use that word?) Bayesian, it’s akin to a mathematician saying that if 2+2 ceases to be 4, that equation goes out the window. I just like what this says about one’s epistemology—we don’t claim to know with dogmatic certainty, but in varying degrees of certainty, which, to bring things full circle, is what Bayes seems to be all about (at least to me, a novice).

More concisely, I like the quote because it draws a line. We can rail against the crazy strict Empiricism that denies rationality, but we won’t hold to a rationality so devoutly that it becomes faith.

• because, coming from a devout (can I use that word?) Bayesian, it’s akin to a mathematician saying that if 2+2 ceases to be 4, that equation goes out the window.

Duhem-Quine is just as much a problem there; from Ludwig Wittgenstein, Remarks on the Foundations of Mathematics:

“If a contradiction were now actually found in arithmetic – that would only prove that an arithmetic with such a contradiction in it could render very good service; and it would be better for us to modify our concept of the certainty required, than to say it would really not yet have been a proper arithmetic.”

Indeed.

To generalize, when we run into skeptical arguments employing the above circularity or fundamental uncertainties, I think of Kripke:

“A skeptical solution of a philosophical problem begins… by conceding that the skeptic’s negative assertions are unanswerable. Nevertheless our ordinary practice or belief is justified because—contrary appearances notwithstanding—it need not require the justification the sceptic has shown to be untenable. And much of the value of the sceptical argument consists precisely in the fact that he has shown that an ordinary practice, if it is to be defended at all, cannot be defended in a certain way.”

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

• I’m not in my comfy chair any more, and I still take the million. Why wouldn’t I?

• Because the million is already there, along with the thousand. Why not get all of it?

• The million isn’t there, because Omega’s simulation was of you confronting Omega, not of you sitting in a comfy chair.

• You aren’t doublethinking hard enough, then.

• I don’t know if this is a joke—I have a poor sense of humour—but you do know Omega predicts your actual behaviour, right? As in, all things taken into account, what you will actually do.

• I am being somewhat … absurd, and on purpose, at that. But I have enough arrogance lying around in my brain to believe that I can trick the super-intelligence.

• Sorry—I’m always inclined to take people on the internet literally. I used to mess with my friends using the same kind of ow-my-brain Prisoner’s-dilemma somersaults, and still I couldn’t recognise a joke.

• That’s alright. My humor, in real life, is based entirely on the fact that only I know I’m joking at the time, and the other person won’t realize it until three days later, when they spontaneously start laughing for no reason they can safely explain. Is that asinine? Yes. Is it hilarious? Hell, yes. So I apologize. I’ll try not to do that.

• Yes. Is it hilarious? Hell, yes.

Not especially, unfortunately. There is something to be said for appearing that you don’t give a @#%! whether other people get your humor in real time but it works best if you care a whole lot about making your humor funny to your audience at the time and then just act like you don’t care about the response you get. Even if people get your joke three days later you still typically end up slightly worse off for the failed transaction.

• Ah. Wrong referent. It’s hilarious for me, and it may, at some point, be hilarious for them. But it’s mostly funny for me. That would be why I took time to mention that it was also, in fact, asinine.

• Because I’d end up with only a thousand, as opposed to a million. And I want the million.

• If Omega has already left, I open box B first, take whatever is in it, and then open box A.

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

• If Omega has already left, I open box B first, take whatever is in it, and then open box A.

Labeling “I decide to lose” as a snark just seems odd.

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.

You are confused. Using Omega is merely a simplification of real possible situations. That is, any situation in which you and the other player have some degree of mutual knowledge. Since those situations are complicated they will sometimes call for cooperation (one boxing, here) but often other considerations or insufficient mutual knowledge will override and call for defection (two boxing).

If you wish to consider the effect of just, say, the mass of a cow then assuming a spherical cow in a vacuum is useful. If the conclusion you reach about the mass of said cow doesn’t suit you and you say “but there are no spherical cows in vacuums!” then you are using an excuse to avoid biting the bullet, not showing your superior awareness of reality.

• You are confused.

Yeah, that’s generally what “I guess my cognition breaks down” means.

If you wish to consider the effect of just, say, the mass of a cow then assuming a spherical cow in a vacuum is useful. If the conclusion you reach about the mass of said cow doesn’t suit you and you say “but there are no spherical cows in vacuums!”

I think you can reasonably expect people to behave in real life as if they expect the laws of physics to approximate reasonably closely what newtonian mechanics predicts about spherical point masses. What I was saying, however, is that you would be wrong to predict that I defect in prisoners’ dilemmas based on my 2-boxing, because for me Newcomb’s problem isn’t connected to those problems for reasons already stated. I hypothesize that I am not alone in that.

• What I was saying, however, is that you would be wrong to predict that I defect in prisoners’ dilemmas based on my 2-boxing, because for me Newcomb’s problem isn’t connected to those problems for reasons already stated. I hypothesize that I am not alone in that.

And I said you are confused regarding this belief and the stated reasons. I don’t doubt that others are confused as well—it’s a rather common response.

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

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

This seems like a pretty symmetric question 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 t