Are you a One-Boxer or a Two-Boxer?

[Jragon]

I’m bored of Monty Hall and .999… = 1. They’ve been beaten to death. Instead I want to open another [del]horrific flame war[/del] calm, logical discussion on logical conundrums. And this one doesn’t even really have a wrong answer (unless you disagree with me :p)! Let’s talk about Newcomb’s Paradox:

You are invited to participate in a strange game show. The game show is run by an enigmatic entity known as The Predictor. The Predictor, as his name suggests, predicts the action you will take during the game. The Predictor entity almost certainly (meaning: 100% of the time or damn close to 100%) accurately predicts your choice – whether through experience, luck, or witchcraft. This is a “don’t fight the hypothetical” point, you have to assume that you’ve been shown sufficient evidence to believe that it’s (nearly) infallible regardless of how unlikely that really is.

The game goes as thus: there are two boxes: A and B. Box A is transparent and always contains $1000. Box B, on the other hand, can contain $1 million or $0. Box B is opaque so you do not know for sure whether money is in the box or not until you open it. You have two choices: you can pick both boxes A and B, or just box B.

Before you take any action, The Predictor predicts whether you’ll pick Box B (One-Boxing) or A and B (Two-Boxing). If it predicts you’ll take both boxes, it will not put money in Box B (because you’re a greedy bastard). If it predicts you’ll only take Box B, it will put the $1 million in the box. After the money is set, you are presented with the boxes, and told to make your choice. You are not aware of the predictor’s prediction (but are aware of his infallibility and the game rules).

So the possibilities are:

Predicts: A & B; You Choose: A & B; Get $1000
Predicts: A & B; You Choose: B; Get $0
Predicts: B; You Choose A & B; Get $1 million + $1 thousand
Predicts B; You Choose B; Get $1 million

Special case: If the predictor thinks that you’ll be tricksy and pick randomly (e.g. flip a coin) there will be no money in Box B.

What the game show has to gain by giving away this much money so easily is left as an exercise to the reader, but assume there’s no malicious intent or anything else unsavory that would throw the facts of the problem into doubt.

My answer is to choose Box B. To be honest, I have trouble seeing the perspective of two-boxers at all. In my mind, the predictor is almost infallible, so if I trust the odds, then I should trust that whatever I pick it predicted. Thus the expected utility of my choices is $1000 for Two-Boxing and $1 million for One-Boxing. Of course, being the attractive, wonderful, clever people you all are I know you’ll agree with me.

Right…

[QUOTE=Wikipedia on this problem]
In his 1969 article, Nozick noted that “To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.”
[/quote]

Have at it!

[ragerdude]

I’m a One-Boxer.

[deltasigma]

So, to get a handle on this, we should assume a sort of entangled state between the contestant and the predictor such that the ‘right’ choice by the former leads to the ‘wrong’ one by the latter and vice-versa?

[Inner_Stickler]

I don’t get it. If it’s a perfect predictor, then the best answer in terms of maximizing money is choosing box B.

[Jragon]

@deltasigma

So, to get a handle on this, we should assume a sort of entangled state between the contestant and the predictor such that the ‘right’ choice by the former leads to the ‘wrong’ one by the latter and vice-versa?

The original formulation of the problem explicitly states that the prediction is not causally linked with the action (either by causing the action to happen, or bizarre reverse-causality). I don’t think this changes my choice much, but direct causal linking is generally considered to eliminate two-boxing as an acceptable action.

Since nobody has two-boxed so far, here’s Wikipedia’s explanation for why you would two-box:

The first strategy argues that, regardless of what prediction the Predictor has made, taking both boxes yields more money. That is, if the prediction is for both A and B to be taken, then the player’s decision becomes a matter of choosing between $1,000 (by taking A and B) and $0 (by taking just B), in which case taking both boxes is obviously preferable. But, even if the prediction is for the player to take only B, then taking both boxes yields $1,001,000, and taking only B yields only $1,000,000—taking both boxes is still better, regardless of which prediction has been made.

There are a lot of two-boxers, arguments over this can get pretty heated, I’m actually astounded that nobody has two-boxed yet.

[Little_Nemo]

I take the one box. The fact that the Predictor has such an outstanding record is enough for me to believe his ability is legit.

People often choose the two boxes though. It’s a form of personal exceptionalism. No matter how much you tell them about the Predictor’s ability to guess everyone’s picks in the past, they’re convincing that their pick is somehow different from all those others and is unpredictable.

[Simplicio]

I think the problem is that the correct answer really depends on the nature of The Predictor. Since there’s no such thing as a Predictor, and such a device is impossible and the question leaves even its actual abilities pretty vague (the OP says: “meaning: 100% of the time or damn close to 100%”, but of course there’s a big difference between “100%” and “damn close to 100%”).

So people divide so strongly on the answer because they interpret The Predictor in different ways. If you assume its actually magic, and will get it right 100% of the time, then it seems obvious you want to choose B, because that will get you a million.

On the other hand, if you just think its a good guesser, and only gets it right “damn close to 100%”, then you choose A&B, because in the real world even a very good guesser can’t determine what you’ll choose with 100% accuracy.

[Jragon]

Well, “damn close to 100%” was meant to convey the concept of almost surely (or “almost certainly”) to people who aren’t familiar with it. It means “as effectively equivalent to 100% as theoretically possible”. In other words, while the chance exists for it to be wrong, by the very nature that failure is a conceivable outcome, it’s so (literally) unmeasurably small that it’s lower than any possible number I could give you.

[Malacandra]

What happens is this:

The Predictor always predicts A & B, because
People reason that there will never be any money in B, so
People always take A & B rather than not get any money, so
The Predictor is 100% correct with his prediction.

This means that the logical choice is two-boxing, and the Predictor gets proved right yet again.

[CandidGamera]

This problem seems very awkwardly constructed, and due to the nature of ‘perfect magical predictor’ would likely collapse into a predictable pattern - once people realize that every previous contestant who has chosen B is a millionaire, and every previous contestant who has chosen A+B is not, over a few dozen ‘episodes’, then everybody would pick B. The show would lose a million dollars per contestant, and people would stop watching.

[RitterSport]

This is a weird problem because the Predictor is benevolent. That is, as long as we’re not greedy, we still get $1 million. Since the difference between $1 million and $1.001 million is so small, I’d rather go with my gut instinct, and the Predictor’s ability, and just choose B. The added utility of the $0.001 million just seems immaterial.

I don’t even see how it’s a question with the numbers as presented.

ETA: posted before I saw the above two responses, but I think CandidGamera states it better.

[chrisk]

Put it this way… you’re never going to get me to admit I’d ever two-box before the Predictor has made his prediction on me.

[Cognoscant]

I think Malacandra is on to something here. There IS an assumption being made when we conclude from the evidence given that “the predictor is almost always right”, but it’s not the simple assumption you’d think.

Let’s take the same scenario: one-box or two-box choice. And before I make my decision, I ask the game show host, “How often did the predictor get it right?” “95% of the time,” assures the game show host. Well those are pretty good odds, I suppose. How many times did the contestant choose two boxes? “95% of the time,” says the host. Hmmm, I think. Now I’ve got to wonder: is the predictor making a real prediction based on individual scenarios and interests? Or is the predictor simply making the same two-box choice every time and leaving it up to the human to “prove” it right?

I suppose what I’m saying is that, given the information supplied in the OP, one-box is the correct choice. However, we don’t have enough information to make it a perfect choice. There could be additional information that we don’t yet know about which could make two-box a better choice.

[marshmallow]

@Jragon

So the possibilities are:

Predicts: A & B; You Choose: A & B; Get $1000
Predicts: A & B; You Choose: B; Get $0
Predicts: B; You Choose A & B; Get $1 million + $1 thousand
Predicts B; You Choose B; Get $1 million

I never got this part of the paradox. If its predictions are wrong it’s not an infallible predictor, by definition. So the bolded can’t happen.

[Jragon]

@CandidGamera

This problem seems very awkwardly constructed, and due to the nature of ‘perfect magical predictor’ would likely collapse into a predictable pattern - once people realize that every previous contestant who has chosen B is a millionaire, and every previous contestant who has chosen A+B is not, over a few dozen ‘episodes’, then everybody would pick B. The show would lose a million dollars per contestant, and people would stop watching.

It’s an alien gameshow, the aliens think of us like puppies and kittens and like to see us happy. Money is no object due to a deal they made with Ronald Reagan. Thus they lose nothing and gain tons of cute video footage via the game show. Is that so wrong?

@marshmallow

I never got this part of the paradox. If its predictions are wrong it’s not an infallible predictor, by definition. So the bolded can’t happen.

It’s because the possibilities are in the conceivable state-space (for lack of a better term), it’s part of the “almost surely” package. The predictor being wrong “almost never” happens (which means 0% chance), but they’re conceivable so it could happen once every few infinities, and thus are listed.

[Jragon]

The 0% is similar to the 0% you get when you work out the probability of an event occurring over a continuous period of time. The probability I’ll turn on my lights exactly one second from now is zero, but at the same time… well, there’s clearly some point where the light switches to “on”.

It’s not exactly the same notion, but it’s similar in principle.

As Wikipedia puts it (bolding mine):

If an event is sure, then it will always happen, and no outcome not in this event can possibly occur. If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0. Thus, one cannot definitively say that these outcomes will never occur, but can for most purposes assume this to be true.

[Bryan_Ekers]

Pick B, don’t spend time analyzing it.

[Bryan_Ekers]

@marshmallow

I never got this part of the paradox. If its predictions are wrong it’s not an infallible predictor, by definition. So the bolded can’t happen.

I gather there’s a very small chance these could happen, or at we’re supposed to accept it as part of the scenario. I figure the alien has predicted my choice before even explaining what the choice is - it’s not in my interest to spend a lot of trying to analyze it, trying to predict what he predicted, so just take the big payout.

At the very least, if I get nothing, I’ll be one of those near-impossible times the alien was wrong. My episode of the alien game show will get many hits on alien youTube (more so if I hem and haw and pretend to be conflicted about the decision I’ve already made - to take B), and maybe I’ll get the home version as a parting gift.

[Ludovic]

I’d pick B because of the utility value of $1000 vs $1 million. $1000 won’t make a difference in my life hardly at all. So there is a good chance that the predictor can physically read my mind enough to know what I will pick such that it makes it worth it to attempt the million.

[MOIDALIZE]

I pick center square for the block.