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 -
My interpretation:
Free will is indistinguishable from randomness, and your brain has some randomness. The alien understands your personality, but can't predict randomness. For example, maybe immediately before the experiment, they cloned you and ran a perceptually identical experiment; then, if your clone picked both boxes in the practice run, the alien didn’t fill the opaque box for the real run.
You can win $1,001,000, but only if you're lucky. For example, let's say you have a 50% chance of choosing both boxes. Then the alien has a 50% chance of filling the opaque box. You have a 25% chance of winning $1,001,000...but a 25% chance of winning $0, and 25% chance of winning $1,000.
You can't trick the alien: if you're more likely to choose both boxes, the alien is less likely to fill the opaque box. Formally, if you with probability p pick both boxes, the alien with probability 1 - p fills the opaque box. Imagine your clone, in the same perceived surroundings, with your same strategy.
If the experiment was repeated ∞ times, on average you'd win $1000p^2 + $1000000(1 - p)^2 + $1001000p(1 - p) = $1000000 - $999000p; increasing p strictly decreases your average win. The statistically optimal strategy is to always pick one box.
Inalterability doesn’t imply futility. Choice and determinism are compatible regardless of how a complete picture of the universe’s physics turn out. All you have to do is rebut the intuition that leads you to fatalism.
Let’s just go ahead and say there’s a means–end relation between a contemplated action and a goal, just in case the desirability of the goal rationally contributes motivation for taking the action which is to say all else being equal (i.e., in the absence of conflicting consequences of higher priority), it makes sense to take the action for the sake of the goal’s achievement.
By what criteria can someone recognize the existence of a particular link? There’s an evidential criterion in that there’s a means–end link from action to goal, just in case the goal is more likely to be found to obtain when the action is found to be taken than when the action is found not to be taken (i.e., the action’s occurrence is correlated with the goal’s occurrence). A counter factual criterion implies there’s a means–end link from action to goal just in case the goal would obtain if the action were taken, but not otherwise (or at least the goal would more likely obtain if the action were taken than if otherwise). A causal criterion would mean there’s a means–end link from action to goal just in case the action causes (or tends to cause) the goal to obtain. And then as already mentioned, a fatalist criterion; which implies there’s never a means–end link from action to goal; and all actions are futile. (No one takes fatalism seriously in practice, but a lot of people believe it would indeed follow if the universe were deterministic; and therefore, they reject determinism).
The second one is somewhat counterintuitive. Inference involves propositions of the form, “If X then Y; we infer consequent Y from antecedent X.” But logic textbooks distinguish many varieties of inference (including subjunctive inference and material implication). Mathematical logic more often uses the latter because it’s much simpler to formalize. In material implications, “If X then Y,” just means, “It is not the case both that X is true and Y is false.”
The first three criteria often coincide with one another. If I take the action of crossing the street to achieve the goal of getting to the other side (and take that to mean that “action” as the initiation of a series of muscle contractions, not as the passage across the street, so the goal’s achievement doesn’t just follow tautologically from the action’s occurrence.) Knowing that I will cross informs you that I’ll get to the other side, but knowing I will not cross informs you otherwise, which justified the evidential criterion. If I walked across the street, I would (likely) get to the other side, but (very likely) not otherwise, fulfilling the counter factual criterion. And my walking across the street ‘causes’ me to get to the other side, fulfilling the causal criterion. By any of those three criteria, there’s a means–end link from the action of crossing to the goal of getting to the other side. Given that means–end link, and other things being equal, my desire to be on the other side rationally motivates my crossing.
The problem in Newcomb’s Paradox though, is that the criteria diverge. Taking just the opaque box, forfeiting the $1,000, is strong evidence that you obtain $1,000,000 in the opaque box, but taking both boxes is strong evidence that the opaque box is empty. But taking the transparent box or not has ‘no’ causal influence on the content of the already-sealed opaque box. The evidential criterion says there’s a means–end link from the action of taking just the opaque box, to the goal of obtaining $1,000,000 in the opaque box #1. But the causal criterion says otherwise; if there’s a means–end link, then it’s an acausal one. (This is also the exact same divergence that happens in real life Prisoner’s Dilemma situations)
The causal and evidential criteria diverge even in some completely mundane cases, where an action correlates with (but doesn’t cause) a subsequent state. If you were to take just the opaque box and not both, then there would be $1,000,000 in the opaque box, #2. In a more rigid formulation, counterfactual links are just causal links, what would differ if you were to take just the opaque box compared with your taking both boxes is whatever taking just the opaque box causes and nothing more than that.
This is why the fatalist criterion can’t be correct. Because there are innumerable means–end links in a deterministic universe. Even under the interpretation of Many-Worlds, quantum mechanics is technically deterministic (in that quantum amplitude flows deterministically through configuration space), and ‘still’ has the property that a given classical state (in some particular configuration-space branch) is followed by a prior, divergent classical successor states (in subsequent branches). It “looks” nondeterministic as far as choice is concerned but in any case present state of our universe is compatible with multiple futures, in some sense or other. Whether or not the multiplicity of futures is genuinely nondeterministic in some sense, the argument still holds. Some degree of multiplicity of futures is compatible with choice (even though it isn’t required), but ‘excessive’ multiplicity would undermine choice.
TL;DR: Take box B.
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No. You are forgetting the correlation. The problem very clearly states that the predictor "almost certainly" will predict your choice. That means that for the 50% that you choose one-box, the predictor won't be filling the mystery box 99.99% of the time. And for the 50% that you chose one-box, the predictor will be filling the mystery box 99.99% of the time.
So the breakdown is: $1,001,000 (0.005%), $1,000,000 (49.995%), $1,000 (49.995%), $0 (0.005%).
That isn't quire right because the predictor is not 100% accurate. If we assume the accuracy is 99.99% (q), then the probability that the predictor will fill the mystery box is (q)(1 - q). Close, but not quite the same.
Correct.
But the real question my essay is trying to explore is why some people do not see that's the case. In my experience the reason why people choose two-boxes is that they completely ignore the accuracy of the predictor, and instead of assuming that q is close to 100%, they simply treat it as a completely unknown variable that could take any value, including 0.01, despite the formulation of the problem.
Why do they do that?
You’re right, I misunderstood the problem.
Why do people gamble?
Alternatively, they also misunderstand the problem. I wonder if the “practice run” method of predicting their behavior would change their mind.
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