Newcomb's Paradox

Great Debates
1

The following is the Newcomb problem as stated by Martin Gardner and Robert Nozick. This problem is currently unresolved.
Two closed boxes, B1 and B2, are on a table. B1 contains $1,000. B2 contains either nothing or $1 million. You do not know which. You have an irrevocable choice between two actions:

  1. Take what is in both boxes.

  2. Take only what is in B2.

Before the test, a superior Being has made a prediction about what you will decide. You have complete confidence in the Being’s ability to predict your actions. He has already accurately predicted your choices many times before in many situations, and you know that, without failing once, he has correctly predicted the choices of millions of people in the very situation that you are now in. You can think of the Being as God, or as a higher intelligence from another planet that took measurements of the state of your brain, thoughts, and surroundings to predict your actions.

If the Being expects you to choose both boxes, he has left B2 empty. If he expects you to take only B2, he has put $1 million in it. If he expects you to randomize your choice, or to use any tricks, he has left B2 empty. In all cases B1 contains $1,000. You understand the situation fully, the Being knows you understand, you know that he knows, and so on.

What should you do?
If you take both boxes, the Being will almost certainly have predicted this and will not have put the $1 million in B2. You will almost certainly get only $1,000. On the other hand, if you take only B2, the Being will almost certainly have predicted this, and the money will be there. You will almost certainly get $1 million. Clearly you should take B2 only.

But the being made his prediction some time ago, and then left. He either put the money in B2 or he didn’t. If you like, you can imagine that the far side of the box is transparent, and others have seen if they money is there or not. Whether the money is there or not, clearly you are better off taking both boxes.
Both arguments seem correct, and yet they reach opposite conclusions.
Personally, I am most drawn to the second argument. The money is there, or it isn’t there. So I should take both boxes.

However, were I to take both boxes, I would fully expect to get only $1,000. Were I to take only B2, I would fully expect to get $1,001,000. The Being has predicted this situation accurately millions of times without fail, and there is no reason to believe he will predict wrong now.

My seemingly contradictory conclusion is that I would definitely take B2 only, all the while still believing that taking both boxes was the “correct decision.” This answer is not satisfying though.

What are the thoughts of the Straight Dope on this problem? Which argument do you believe (or do you have a third argument)? What would you do? Let’s put this paradox to rest, if possible.

2

Huh? Why would you get the extra thousand if you took only B2?

I’d take both boxes: anybody who knows me would expect me to try and figure out some way to beat the game (which means the predictor would leave B2 empty).

3

I’d flip a coin; there goes your “you can always predict what I’ll choose”.

4

What if they could predict the subtle motions of your thumb and the local air currents? :wink:

5

The OP says that if the Being predicted you’d try and randomize your choice, he’d leave the box empty.

I’d choose the box closest to me, since Because Superior Beings comes from Australia, as everyone knows, and Australia is entirely peopled with criminals, and criminals are used to having people not trust them, so he’d try and trick me by putting the million in the box closest to me.

6

Ok, to face the dilemma straight on then:

It seems to me that in a sense the scenerio is misleading. If the entity can perfectly calculate whatever you’ll do given all factors, then to speak of your “choosing” one option or the other is false. Your actions would have to be completely deterministic, and the entity would know the outcome of all possible factors, including his giving you a “choice”. The only possible source of paradox would be if the setup created an infinite recursive loop similar to wiring a light and a photoreceptor together so it constantly oscillates. The question then would be predicting what additional factors might dampen out the oscillation and whether the final resting state would be yes or no.

7

I dunno why this is considered such a head-scratcher. It’s no different from any scenario in which someone makes a prediction and the prediction itself has a chance to influence the outcome.

8

I don’t get the problem with it either. It seems more like it’s really meant to question whether you believe in predictive powers.

The apparent difficulty only seems to arise if you confuse the context of the problem (in which there is a perfect predictor and causality does not apply as we know it) with ordinary experience (in which we do not expect a perfect prediction to occur).

The obvious flaw is in the statement “Either the money is in Box B or it isn’t.” If you assume a ‘perfect predictor’ then the fact of the money being placed was not determined arbitrarily beforehand as this implies.

Even if you don’t believe in a totally perfect predictor, imagine that you have some confidence in the ability of the predictor to guess. If the prediction is only slightly better than completely random, you’re better off with Box B.

9

You’ve forgotten two key components: the boxes are see-through, and selecting boxes carries a cost of $1. The big deal is that an omniscient being will be wrong, which is categorically impossible for an omniscient being–hence, the concept of omniscience is logically incoherent. As my friend Corey stated, it is “as profound as it is obtuse.”

10

I don’t get it. If you pick B1, you are guaranteed $1000, but you automatically don’t get $1 million, whether it is there or not, this isn’t a very good choice.

If you pick B2, you are almost guaranteed $1 million. If the predictor is wrong, you lose only $1000 of the possible money you would have had anyway.

If you choose both boxes you are almost guaranteed to lose $1 million, and you are betting on the very small chance that a predictor that has never been wrong before, is wrong about this one instance.

Seems to me the obvious choice is to pick B2. A virtually guaranteed $1,000,000 versus an almost impossible $1,001,000 (because there’s greedy and then there’s stupid greedy.)

11

To me, the problem seems rather stupid.

  1. There is an all-knowing, prescient Predictor
  2. But you’re too stupid to believe us so you should consider that maybe he isn’t

The only paradox in the paradox is it telling you to pretend that it is one. If you accept everything as true that it states as a truth, and ignore the “but pretend we’re lying”, then there’s really nothing particularly difficult, let alone paradoxical, in the whole thing.

You are required to try and maximise your outcome and the Predictor is required to keep you from getting it all. There’s no other choice than taking the second largest possible value. By definition of “prescience” there’s just plain off no way to catch the Predictor off-guard.

12

Calvinist, huh?

I agree. Unless I misread something, the obvious answer is to pick B2. This should be obvious to the supreme being, therefore guaranteeing $1mil.

13

You (and several others) are adding a lot to the problem that wasn’t there.

The problem does not assume an all-knowing, prescient predictor.

We have confidence that the predictor will be correct because his past success justifies such confidence. He has previously predicted our actions, and predicted the choices of others in the same situation.

It has been argued that the existence of a perfect predictor negates free will.

Does it not also follow that the impossibility of a good (not perfect) predictor implies randomness?

Further, does pure randomness itself even allow for free will? I can hardly be said to have chosen something if what I did was totally random.

The only conclusion that does not negate free will is that a very good (but not perfect) predictor is possible, which is precisely the context of the problem.
In that setting, however, the problem remains. I still can think of no way to counter the essential fact that the money is already either there or not there by the time I make my choice. But at the same time, anyone watching would bet a large sum that if I take B2 only I will get the million, and if I take both I will get only the thousand. And they would almost certainly be right.

14

I don’t see the paradox in the OP. Any thoughts of “oh, there must be money in B2, so I can take both” would have been predicted, so taking both would be foolish and result in losing $999,000. The fact that others can see how much loot you’re going to get before you do doesn’t make any difference that I can tell.

ForumBot’s version adds in a li’l bit of paradox, unless you consider playing the game for anything other than maximum return to be a “trick”, in which case you would only ever see $1000 in B1, meaning the omniscient being either correctly predicted you’d take both boxes, or he correctly predicted you’d be a jerkoff and not play the game for maximum return.

15

Just because you can negate free will doesn’t make it a paradox. It just means that you described a literary fiction where there isn’t free will.

No, because the logic of the situation requires a a single answer. Just because the predictor could be wrong doesn’t mean that there is some chance that he will eventually choose the wrong answer.

It’s like saying “Tell me the answer to 2 + 2” but then assuming that there’s some chance that I’ll say 78 instead of 4. Yes, in the real world there’s some chance that someone will say some random number just to be a prick, but the real world is divorced from a game theory world. In a game theory world it’s a basic assumption that both parties are trying to maximise something and doing anything else would be stupid. And so the answer will always come back as 4.

So both by inductive reasoning and self-fulfilling prophecy, the Predictor will always assume that the person will choose B2, and the person will assume that the Predictor knew he was going to do that.

That doesn’t negate free will. It just assumes that both parties are rational and have no reason to trust that the other party will fail.

16

But the money either is, or is not, in the box. Letting other people look in through the transparent back only makes that more obvious.

Your choice will not make the money vanish if it is there, nor make it appear if it isn’t.

So clearly the best strategy is to take both boxes.
I am totally convinced by this argument, and yet I would still only take B2. Because I would rather have the money than make the “correct choice” and taking B2 by itself means I will almost certainly get the money.

Still, how can the other strategy be wrong? The money is either there or not.

17

You are underestimating the other argument.

In fact, when this problem has been posed, about 30% of people say they would take both boxes.

It is hardly a foregone conclusion that everyone would go for B2 only.

18

So? You keep repeating this, but I fail to see what relevance it has. As long as the predictor has a better than random track record, the obvious choice is B2.

How is this different from, say, betting on a horse race after the race has been run in secret and no results released? Either horse A won or horse B won, and my betting isn’t going to change the outcome, but if I know of an expert with a good track record I’m going to listen to him.

19

But the money is either there, or it is not there.

In either case, taking both boxes maximizes your earnings.

In terms of game theory, taking both boxes is dominant.

But the expected utility of taking only B2 is much higher.

20

No it isn’t. Let’s give the predictor a 99% chance of being right, for argument’s sake. That means that by taking both boxes, you expect to get 1000+(0.011000000), which is 11000. By taking B2, you expect to get 0.991000000, which is 990000. 11000<990000.

Since the predictor has never been wrong, his chance may be greater than 99%, which makes it even more lopsided.

EDIT: If the predictor’s chance drops to 50.05%, the expected value of each choice is equal. If he is even a little bit better than that, B2 is the correct choice.