Newcomb's problem splits people 50/50 in two camps, but the interesting thing is that both sides think the answer is obvious, and both sides think the other side is being silly. When I created a video criticizing Veritasium's video This Paradox Splits Smart People 50/50 I received a ton of feedback particularly from the two-box camp and I simply could not convince anyone of why they were wrong.
That lead me to believe there must be some cognitive trap at play: someone must be not seeing something clearly. After a ton of debates, reading the literature, considering similar problems, discussing with LLMs, and just thinking deeply, I believe the core of the problem is recursive thinking.
Some people are fluent in recursivity, and for them certain kind of problems are obvious, but not everyone thinks the same way.
My essay touches Newcomb's problem, but the real focus is on why some people are predisposed to a certain choice, and I contend free will, determinism, and the sense of self, all affect Newcomb's problem and recursivity fluency predisposes certain views, in particular a proper understanding of embedded agency must predispose a particular (correct) choice.
I do not see how any of this is not obvious, but that's part of the problem, because that's likely due to my prior commitments not being the same as the ones of people who pick two-boxes. But I would like to hear if any two-boxer can point out any flaw in my reasoning.
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Notes -
The problem with Newcomb's problem is that it basically involves time travel, and generally underspecifies how that time travel works. Consider a similar problem:
Time 1: you discover box 1 with 1,000,000 points
Time 2: you discover a box 2 with 1,000 points
Time 3: someone shows up claiming to be a time traveler shows up and says that if you hand him box 2, he will multiply it by 1,000, and then go back in time to put it in box 1. Actually, he claims, that's where box 1 came from all along and if you don't give him the 1,000 your box 1 will disappear.
Assuming you are rational/selfish, whether you say yes or no very much depends on whether he's telling the truth. If the problem carefully specifies that he actually is a time traveler telling the truth, and time travel does work this way, then obviously you should give it to him (one box). If this happened in real life, I would not give him anything and two box, because my prior on time travel existing is less than 1/1000 and he's just a liar trying to con me. If the problem is not careful and is ambiguous about his truthfulness then people's answers are going to depend on their trustfulness, suspension of disbelief, or just general attitudes towards how willing they are to buy time travel in a hypothetical logic puzzle.
Actual Newcomb's problem is basically the same as this in that decisions you make in the future affect things in the past, and the being making the boxes has to have time travel powers in order to guarantee a 100% success rate (though not all version of the problem specify this precisely, maybe it just has a 99% success rate, or a vague but high success rate) The reason people so confidently disagree is that in any well-specified version of the problem the answer is obvious, but in any vague under-specification it's ambiguous to which well-specified version people will round it to. This is the exact same reason the Monty Hall problem is controversial as well. It's not merely there being a counter-intuitive answer, it's that the problem specifications are very volatile and people keep leaving important details ambiguous that they shouldn't.
Not time travel, just perfect prediction. If you're actually a perfect predictor then you can in essence see the future. If you had a perfect model of physics and initial conditions then you could predict a coin flip with 100% accuracy. The kind of reason a human does when presented with the boxes is no different unless you a proposing some spooky non-material stuff in the reasoning. The formulation I'm familiar with is perfect prediction in which case there are four theoretical cases.
You one box and Omega correctly predicted you would one box thus you get $1m
You one box and omega incorrectly predicted you would 2 box so you get zero. This is impossible by construction, omega cannot predict wrongly.
You two box and Omega incorrectly assumed you would one box, you get $1m + $1k. This is impossible by construction, Omega cannot be wrong.
You two box and Omega correctly guesses you'll two box. You get $1k.
There are only two actually possible options with the given constraints and you get to make a choice which of them is the case. This is not a paradox unless predicting future events is impossible.
Your whole reasoning relies on there being something intrinsically impossible about predicting your decisions, even as you lay out the reasoning for them. Is it so hard to imagine that someone could read you well enough to know which outcome you'll ultimately reach?
Corporate want's you to find the difference between these two pictures and they are the same picture. Either they KNOW the future or they don't, and the problem as typically presented gives us no reason to believe that they do.
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This is only possible if you model people as deterministic mechanisms and not as rational game theory agents. If Newcomb's problem posits that you make a decision AFTER Omega makes a decision, then Omega can be wrong. For instance, if you play a mixed strategy (one box with 50% probability, two box with 50% probability), then omega has to not only model your brain perfectly, but also your coin flip. If you used quantum decay to randomize then it would have to predict that perfectly. If omega can perfectly predict that then you've removed an important tool in Game Theory. It's like trying to argue that Rock Paper Scissors as a paradox because no matter what you do your opponent can predict you and defeat you. If you tell me that I can't do mixed strategies, or that other people can, in the past, respond to my mixed strategy outcomes then that's fundamentally incompatible with 90% of Game Theory. I don't even know what the rules of this are or what the goal is. Omega can do literally whatever it wants and I'll get literally any payoff that it chooses to give me. I suppose if it's a god that wants to punish two boxers then I guess I'll obey its commandments and one box, but that's not adjency, that's not game theory, that's just submitting to the religious edicts of a higher power with arbitrary rules.
We can model it much more simply by making an alternate version where, at time 1, you decide to one box or two box. Then at time 2 omega is informed of your decision and puts stuff in a box, then at time 3 you get the result of the decision you already made. Here you obviously 1 box. This is a very straightforward, simple, and uninteresting game theory problem. The problem with this then is not what the original Newcomb's box says happens. It says you make the decision after omega does. If you actually mean that people make the decision before omega then saying they make it after is lying.
Maybe this is still useful as a critique of attempts to map Game Theory to reality. Essentially saying "Every game is an iterated game played out over the course of your lifetime. Any decisions you make will affect your personality and reputation, so doing greedy things will hurt you in the long term even if they are the rationally correct choice to a one shot game that you see in the short term." Which, sure. This is how cooperation can exist in prisoner's dilemma-like situations, because you cannot incentivize (and it is irrational to try) to cooperate in a true, pure, one-shot prisoner's dilemma with no modifications, but none of those conditions apply in real life. But you also don't have mind-reading omegas in real life either, so I don't think that's quite what people mean by this.
Ultimately, the premises are fundamentally contradictory, so the only way to come to a solution is to suspend disbelief on half of them. Either you believe that omega can perfectly predict you and you have no agency, so hope that your were born as a one-boxer (because you don't get to decide), or you believe you are a rational agent who can make decisions when it says you can, in which case omega can't predict you so might as well two box. But these are beliefs about the premises of the problem, not about what is good epistemic or rational behavior in a given coherent scenario (which always follows logically and mathematically from the premises and math to determine the maximum payoff)
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No it doesn't. That's what two-boxers claim in order to fit the problem into their view of reality, but that commits an appeal to incredulity fallacy.
That's not the Newcomb's problem: 100% success rate is never specified, it's "almost certainly". That means close to 100%, not 100%.
You are saying it's not possible for Omega to have such accuracy unless the future affects the past, but you don't provide any justification for that. You are basically saying: "I don't see how X is possible, therefore X is not possible". That is not a valid argument, that's an appeal to incredulity fallacy.
That is precisely why I devised my sunscreen problem. Your argument is the same as saying: "I don't see how efficacy against skin cancer and eating ice cream could be causally related, therefore they are not causally related".
Just because you don't see how Omega could predict your choice almost certainly without backwards causality doesn't mean that it can't.
This is why I complain about it being underspecified. If omega can be wrong then the entirety of the problem hinges on when/how/why it can be wrong. If it's possible for someone to get away with two boxing and get both boxes, and you can put yourself in that scenario, then you can win by two boxing. If omega attempts to minimize its failed prediction rate, maybe you can employ a mixed strategy (flip a very slightly weighted coin) which randomizes and then you could one box with probability 50.01% and two box with probability 49.99%, causing omega to predict you will one box, and you always get the one box plus almost half the time you get a bonus. Can it predict coin tosses before they're made? Can it predict radioactive decays? This is not mere psychology. I'm not saying it's impossible for someone to cold read you and make educated guesses. If I read psychological profiles on people I could guess that sneakier, greedier, more disagreeable people are more likely to two box while straightforward, naive, or chill people are more likely to one box, and probably get like a 70-80% success rate. Is that what omega is doing? Because then I'm just screwed: I overthink things and seem like a two boxer and if I bit the bullet and decided to one box I would end up getting nothing because it would false guess me as a two boxer.
Literally none of this is explained in the premise. The problem very much depends on information is not present. If I give you "MathWizard's Paradox" and say
"There are two boxes. The left box has some money. The right box has a different amount of money not equal to the left box. You only get one box, which one do you pick?"
This likewise is going to lead to disagreement (or would, if people cared and tried to argue about it). If I added a whole bunch of window dressing to disguise the obvious stupidity of this problem, a bunch of superficial characteristics that made it seem more interesting and less obvious, it wouldn't change the underlying symmetry and lack of information. I have, in my head, decided how much money is in each box. There is a correct answer. But I haven't told you enough information for you to deduce it, and there are infinite variations of this problem, half of which have the opposite correct answer.
It's not that I can't see a way for this to happen, it's that I can imagine a dozen hypothetical ways it could try to do this, and half of them let me two box anyway while half of them don't.
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