Nuances of Newcomb's Problem

; related : -

• Functional Decision Theory

Introduction

First, let me reintroduce Newcomb's Problem.

For this thought experiment, imagine there is a predictor capable of observing human behavior and correctly deducing how one will behave in the future. Maybe this predictor has psychic powers, or is really good at reading body language. I often see it described as a superintelligent AI system. More on that later.

This predictor already knows enough about you and has already predicted what answer you'll give the following problem. Having made its judgment, it places down two cardboard boxes. You get to chose which ones to open, and you keep whatever is inside.

The box on the right always contains a thousand dollars. The box on the left contains a million dollars, but only if it predicted you'd chose only that box.

Once it explains the situation, it walks away. It can no longer affect whats inside the box, the choice is yours.

All that matters is whether you open the one box on the left. (If you open the right box, you might as well open the left.) Thus, the two possible answers you can give are "one box" or "two box."

This is a very contentious decision problem. Philosophers couldn't decide on the right answer (and still can't). It's been decades, and a lot of different theories have been put forth. The mainstream falls into roughly two camps:

One camp argues: the predictor has already left, so the left box either has the money or doesn't. If it has the money, you'd be better off picking both. If it doesn't have the money, you'd be better off picking both. Therefore one should pick both.

Unfortunately, this sort of person only gets a thousand dollars, by premise.

Which sucks, but what can you do? It isn't your fault for being rational! We have created a question that punishes reasoning correctly, and reasoning correctly gives the wrong answer, surprised pikachu face. Perfect predictors don't actually exist in real life, who cares.

The other camp argues: everyone who picks one box gets a million dollars, and everyone who picks two boxes gets a thousand dollars. Picking one box is evidence of a better outcome, so I pick one box.

And it has a million dollars!

The first camp sees this, and says "you idiot! if you had picked both boxes, you'd be a thousand dollars richer right now!"

And they reply: "if i had picked both boxes, I wouldn't have the million," and they proceed to laugh all the way to the bank.

But it gets weirder than this. Consider a modified scenario: instead of cardboard, the boxes are made out of transparent plastic. You can plainly see whether the other box has a million or not.

This doesn't change the first camp's answer, though they'll probably not bother opening the empty plastic bin (which is all they'll ever see.)

But I'm pretty sure the second camp will continue to only open one box, leaving the thousand dollars sitting there (no one ever got both) but I'm not actually 100% sure. TBH, I don't think about the second camp often.

But why not? you might think those girls are on to something, if they can get the right answer here.

But consider a modified scenario. The predictor is in fact a fraud. Sure, they're right most of the time, but after thousands of experiments, a researcher figured out the trick being used to predict people. You see, the predictor first collects a stray hair and sequences your DNA. It turns out there's a strong correlation between certain genes and philosophical inclinations, so this procedure is pretty effective at predicting people's answers.

Of course, this explanation is all well and good, but you don't know if you have the two-boxing gene or not. How would knowing the mechanism change your decision?

The first camp continue to pick two boxes, of course. Whether the second camp picks one depends on how tight the correlation is. If it's greater than around 99.9%, then they know one boxers get more money on average, so it's better evidence of winning big than knowing they pick two.

(If you're really into the weeds of these arguments, you may recognize the above scenario as one that's usually presented in a more confusing way, by imagining a world where smoking doesn't cause cancer.)

Anyway I think there's an important difference between these two problems. The best way to highlight the difference is to ask what you would do if you had heard some predictor was going around springing this problem on people before the predictor learned anything about you.

The first camp, in the first problem only, would reason as like this: if there's someone who can predict my behavior, it's best for me if they predict I'll pick one box.

One extreme way to accomplish this by giving some guy a gun and paying him to shoot you if you don't pick one box. Now when you meet the predictor, it will predict that you will be afraid of getting shoot and chose one box, but since it only cares what you chose, it'll stash the million and walk away. Yet now if you try to two box, you get shot, so you one box still.

In general, this is something called a precomittment. It's like the captain who ties himself to the mast so he can hear the sirens without being tempted to climb overboard. The details of how you ensure you're committed aren't that important, as long as they ensure you'll make a certain decision even when you later want to chose differently.

But consider what you'd do in the second scenario. If the predictor is scanning your DNA for a certain gene, then you could scan your dna first and see if you have the gene. Armed with this knowledge, you can chose two boxes even if you have the one-box gene, because you know it's not a perfect predictor (the gene influences but doesn't determine your choice), and any lucky outlier who has the gene and two boxed anyway can get both piles of money!

Importantly, in the second scenario, there's no sense in "precommiting" to take one-box, not if the prediction is already determined by your genes.

Subjunctive Dependence

So there seems to be a meaningful difference between "true prediction" and statistical correlations that might be wrong. But it's not enough that the predictor is sometimes wrong, because consider what happens if we modified the scenario so that the predictor remains (technically) perfect, but to make things slightly interesting, it also rolls dice so that 0.1% of the time, it ignores its own prediction and lets the dice decide whether to put the million in or not. Thus, we get'd the same statistics as the genetic version: strong but imperfect correlation.

And this doesn't change anything! The first camp would still take the commitment to one-box because an almost-certain chance of a million is better than one thousand and an almost-negligible chance of a million.

If a tree falls in a forest and no one is around to hear it, does it still make a sound? The truth is that this question doesn't make sense. The tree creates physics of sound, but not perception. It's pure semantics: you have a gut feeling that one or the other is the definition of sound, and the resulting dispute is a merely verbal disagreement.

A similar semantics problem lurks in Newcomb's Problem.

Consider the following explanation of its prediction capabilities: you are currently inside the matrix. This world is just a computer program, a simulation that can be rewinded or restarted. You have no idea if you're in the matrix or the real world. Maybe there isn't a real world.

One simulation of you is presented the two boxes and asked to decide. The simulation ends once you've made your choice. Based on this data, the predictor poses the same problem to another version of you. Maybe it's the real you, maybe it's another simulation but one which won't end prematurely.

The point is, with this framing, I think the first camp would treat the scenario differently. Because it makes clear what's happening: you're deciding how the simulation acts, and you're deciding how the real you acts, and you can't tell the difference, by premise.

Now imagine right before you make your decision, the predictive AI says, "Oh, you're the real version BTW." The simulation was not told this, so now the AI's data can't actually determine how this might changes your behavior.

Out of curiosity, the first camp might stop and try to guess how the simulated copy might've acted, but it doesn't matter, what's done is done, and now they can reap all the rewards.

(Another way to get the same results: imagine you are a demon who takes possession of the test subject right before they make their decision, and the predictor can't predict demons.)

Point is, this is the actual question the first camp thinks they were answering all along.

It only makes sense to reason as if the prediction "already happened" if you have ironclad certainty you're not in the matrix right now.

But if you don't like the idea of simulated copies and the identity questions that raises, there's a bunch of others routes to the same insight. Imagine the predictor is actually a time traveler, and whenever it "mispredicts", it goes back in time to try a different guess. Imagine the predictor is actually a hypnotist, able to put you into a trance where you can forget any memory. Once you chose, your memory gets erased, and you chose again.

The point is, in these scenarios, it's clear that it hasn't already happened, and your choice might actually change whether the "real you" gets a favorable or unfavorable situation. So you pick the choice that's favorable in both scenarios, which is one boxing.

If the predictor is unreliable, then you still aren't certain you aren't in the matrix. If it's 75% accurate, then it's indistinguishable from having been in in the matrix 37.5% of the time. (or 25% of the time, the time travel or hypnotism fails, etc.)

Remember how I said there were two camps? There's three actually.

A third group of weirdos, who aren't taken very seriously by academic philosoph