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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Your choice reveals what kind of person you are, which omega already knew. If you didn't know what you were going to choose ahead of time that's a mark of your ignorance, not omega's.
I.e. my choice doesn't change anything, it just "reveals" information already known to the relevant player Omega.
What I know or don't know ahead of time doesn't matter, because I'm not making a decision ahead of time.
You can also just model it as omega knowing whether or not you're smart or lucky enough to come up with the right answer to get the $1m. If you pick the right answer you get $1m if you don't then you don't. It's a bit of a brain twister but it works out.
But that's kinda circular, because whether the answer does get me the money depends on Omega's knowledge and decision. So whether I'm the kind of person who Omega rewards is luck, and my decision doesn't retroactively affect it.
In a clockwork universe it's of course all luck all the way down.
In a clockwork universe, is there such a thing as decision theory, or a subset thereof known as game theory? It would seem to me that, sure, one could have a mathematical theory of optimization, extremal values, or even min-max theory, but it would not seem to me that one could view any such results as being prescriptive - I.e., "If you are trying to accomplish X, you should choose Y." Instead, it would simply observe, "You might by chance (or deterministic integration of physical differential equations or whatever) take action Y or Y', and it turns out that we can compute that Y is optimal for purpose X, while Y' is suboptimal."
That is, if one is an adherent to this conception of a clockwork universe, I think the way they would state their position on Newcomb's problem would be something more like, "You will either 1-box or 2-box, based on the movements of the clock. We can also compute from axioms regarding the clock's movements that 1-boxers will possess more money," and less like, "You're in this hypothetical situation where you need to think about the rational way to proceed optimally, and here is why you should choose to act in the following way." I think if such proponents presented their perspective in this way, it would be less amenable to criticism that their problem is ill-posed as a decision/game theory problem.
The axioms of decision/game theory seem to conflict with the axioms that seem to appear here. I guess one way to put words to that would be that one has a feasible action set within the underlying dynamical system that has cardinality greater than one. Perhaps another way to put it is that it does not seem to me that decision/game theory is applicable to clocks. The feasible action set of clocks has cardinality one. One does not ask how a clock should choose among non-identical actions, though one may observe whether a clock's deterministic actions are/are not optimal according to some metric.
Taking this alternative position would, I think, sidestep the criticism I relayed above from Wolpert/Benford, as what they were fundamentally trying to do was to formalize the problem within decision/game theory, where players have feasible action sets with cardinality larger than one. They observed that if you do this, you run into contradictions without further specification. But it would seem like, sure, if you give up on that, give up on saying that it has anything to do with decision/game theory, that it's more like just making an observation about clocks and optimality/suboptimality, then I think you do avoid the critique.
This seems more or less yud's position.
This seems basically identical. What you decide to do is determined but all of the reasoning is part of how it happens anyways. From the first moment time one boxers were always going to follow this game theory reasoning and none the less, like how a domino still does cause the next domino to fall, the reasoning is still relevant. I just don't really see what the game theory problem here is, choices are relevant to the reasoning that individuals do, free will or not that's how the meat computer in or head picks which muscles to contract.
I mean, kinda no? That's where the Wolpert/Benford critique comes in. You can't formalize the problem in terms of game theory without adding additional assumptions. If your additional assumptions to formalize it are, "It's actually a clock, and there's no feasible action set with cardinality greater than one," then sure, you have a suitable formalization... but it's kinda not game theory. If you want to back away from that being your additional assumption, it's kinda still on you to state other formal additional assumptions that make it a well-posed game.
EDIT: Perhaps another way of describing it would be as follows. Suppose one is just analyzing a clock. We'll discretize time for now just to make it simple. Say that we observe from our analysis that in the transition from time t_1 to t_2, the clock will become one second slow compared to some 'objective' time (handwave any difficulties here). We could observe that this is, in some sense, suboptimal, sure.
Now, does it make sense to say something like, "What if we just call this suboptimal action y' and hypothesize an alternative action y that doesn't result in being one second slow?" Would it make sense to say that we have constructed a decision theory problem? Note that we're not specifying anything about any sort of real policy space or anything; it's not like we're saying, "Here is the policy space of possible mechanisms that a non-clock can choose from to design the clock."1 We just have a clock.
Suppose we say that there is some being, Omega, who will accurately predict that said clock will take action y' and become one second slow, and then put some quantity of money in front of the clock. Suppose we say, "Well, imagine the clock took hypothetical action y, which it can't do, then imagine that Omega would put a different quantity of money in front of the clock in that case." Does this become a game theory problem? If so, what am I supposed to solve for? What is the space of possible solutions?
1 - This is perhaps related to my comment about what Yud did to the prisoner's dilemma problem. He created some different policy space about source codes.
Think of a fully clockwork universe, down to the apparently random quantum phenomena. IFF you have the full picture, can see the entanglement at the big bang, can calculate out the impossibly complex gear movements, you can know what a 'random' quantum event is going to be. But from the limited perspective, inside a finite light cone, you cannot. The initial entanglement is invisible, some of the gears are hidden. This doesn't make it not clockwork, it just means you don't have access and so quantum randomness is the best you can do.
We have the same thing here - from our perspective, lacking whatever it is that gives Omega the accuracy, whatever lets it see the gears we can't, our choice looks free and the answer uncorrelated. It is correlated, which is why Omega can be as accurate as it is, but we can't see the same thing. So ordinarily, we use our game theory and our strict domination and it works out fine. But here, we have a cheat. The problem has given us information we otherwise wouldn't have access to - we now know there is, in fact, a correlation between our 'future' choice and the contents of the box now. We don't know how it works, we don't know why it works, but we have been given that information. We can cheat the problem by leveraging that information. That's why one boxing is rational. That's why ordinary game theory goes away. We have something stronger to use in our thinking.
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