Brainteaser: The Newcomb Puzzle

[frogus]

A mysterious being has come to Earth. It is called Newcomb, and it claims to have predictive powers...it says it can see the future. It has made millions of predictions for various governments about things as diverse as people's golf averages and un-predictably terminating computer programs for the complicated tasks in number theory, and it's predictions have been correct 90% of the time in the past.

Newcomb has set up a game, with the massive quantities of money it has gained by its predictive powers, and millions of people have already played the game, and thousands are going too play every hour. This is the game:

Newcomb has 2 boxes, A and B, and he always puts £1000 in box B, always. In box A, he may either put £1000,000 or nothing, and all one has to do when playing the game is to decide whether to take the money from both boxes A and B, or just take the money in box A.

However, Newcomb puts the money in each box before you play the game. When you come to him there is no way to change what's in them. The way he decides what to put in each box is this:

He predicts whether you will chose both boxes A and B, or just box A. If he predicts that you will pick box A only, he will fill it with the £1000,000, and if he predicts that you will take both boxes, he will leave box A empty.

Will you take boxes A and B, or just box A?

please answer even if you can't figure out the maths behind this... It is a philosophical question as much as logical, and it is interesting to see what people would do when confronted with those two big boxes, and forced to make a decision.

[Kameleon]

...it seems obvious what the logical course of action is. If you choose to take both boxes, there is a 10% chance that you will get £2000 and a 90% chance that you will get £1000. If you choose to take only box A, there is a 90% chance that you get £1000 and a 10% chance that you get zippo. So unless I'm missing something, taking both boxes will give you a chance at lots of money, while taking only box A will mean there is no chance that you get £2000, and even a chance that you will come out empty-handed.

I'd take both boxes.

[Aegis]

Well, I would kick him in the balls, take his wallet, and both boxes.

Actually, I'd take one box. Gaurenteed reward.

[Ned Flanders]

What would you amphibians spend the legal tender on anyway. I'll side with kameleon's reasoning. Why take a risk at getting nothing when you know there is a safe option of some gains. Furthermore, the safe option also contains the element of chance of walking away with the biggest payload. Perhaps I/we are missing something here.

[Kameleon]

Quote:

Originally posted by Ned Flanders Perhaps I/we are missing something here.

Thats what I thought...

[Demis]

I think that the second amount is 1 000 000 and not $1000.

If this is the case, either you choose A&B box for a sure $1000 and a chance of 10% to get the mil (safe way), or choose one box (you don't know if is A or B) and get $1000 if you chose B or a 90% chance to get the mil if you choose A(risk way).

Choosing one box only a chance of 5% to get nothing so i would choose one box.

[frogus]

you have both read the question wrong I think...anyway, to summarize, if you are absolutely certain that Newcomb really has got predictive powers then if:

You chose Box A only:
There is a 90% chance that you will get £1000,000, and a 10% chance that you will get nothing whatsoever. Nada...

You chose Boxes A and B:
There is a 100% chance that you will get £1000, and a 10%chance that you will get £1001,000.

but what does 90% chance mean? There either is or there isn't money in box A.
It isn't really a question of maths though, and I'll tell you why:

Remember that there is no way that the ammount in those boxes can be changed now. It's all down to you. So if you do decide to go for just box A (which it seems Kameleon ought to do -it has a much better statistical pay-off) then what's to stop you just saying 'No! I want both!'. The money's not gonna change...and why should you not just reap the rewards of his prediction?

[Jace]

I think that Kameleon got the figures wrong.

It is £10 Million or nothing in box A and £1,000 in box B

So that is 10% chance of £10,001,000 and 90% of £1,000
or 90% chance or £10,000,000 and 10% chance of Nothing

The odds say to go for box A only and risk getting nothing,
or to go for both and almost certainly getting little.

I'd take the risk and go for A.

Edit - sorry Froggy, we both got there together.

[Jace]

Sorry for the double post

This seems to be a bit of a Schrodingers(?) cat problem.
Is the money there or is it not?
You can't tell until after you make your choice.
Changing your mind does not come into it, you have not chosen until the box is open, once the box is open it is too late to change your mind.

[Ned Flanders]

I read the pound values wrong as well.

[frogus]

now we're getting somewhere Jace! What's Schrodinger's Cat? Anyway, this one runs deeper than it seems...there are several interesting paradoxes in it, to do with free will. Anyway, the fact that you chose A says a lot about your personality traits, however, you're just saying that you'd take A, we can't tell what you'd actually do in the situation of Newcomb landing, but it's interesting think about anyway...

[CM]

I would take both, better safe than sorry!

[Jace]

Schrodingers Cat is an alegory used in Quantum Machanics. Basicly it says that if you put a cat in a box and leave it there for long enough it will die. If you open the box early enough it will be still alive. The problem comes when you try to predict if the cat is alive or not. There is no way of telling until you open the box.

This was used to try to explain some of the paradoxical behaviour of electrons seeming to exsist or not and having no way of predicting it.

With out knowing how Newcomb makes the predictions it is not possible to say anything about free will. He could just be very good at reading trends and bodylanguage.

If I trusted that Newcomb could predict as stated then I would Choose A as the odds favour a big return.

[Baldursgate Fan]

Aah, at last! Someone with a philosophical bent! Thank you Frogus!

On the one hand, the evidence is fairly obvious that if you choose to take only Box A, you will stand a chance to get one million pounds, whereas if you take both boxes you get only a miserable thousand. Taking both boxes will involve considerable risk with the reward being a thousand pounds more.

On the other hand, at the time you make your decision, Box A is already empty or else contains a million pounds. Either way, if you take both boxes you get a thousand dollars more than if you take the Box A only.

There are two accepted principles of decision theory in conflict here. The expected-utility principle (based on the probability of each outcome) argues that you should take A only.

The dominance principle, however, says that if one strategy is always better, no matter what the circumstances, then you should pick it. And no matter what A contains, you are $1000 richer if you take both boxes than if you take A only.

Some of the suggested solutions that we have seen are attempts to show/imply that the situation as presented cannot occur. For instance, some say that it is impossible to predict human behavior with this kind of accuracy. That may very well be true, but even if it is physically impossible, that is not a satisfactory solution to a logical problem. Provided it is logically possible, we are still faced with a puzzle.

One possible solution is the following: The dominance principle is not valid in the puzzle because the statements ("1 million is placed in the box" and "nothing is placed in the box") are not probabilistically independent of the actions ("take both" and "take only A"). The dominance principle is acceptable only if the statements are probabilistically independent of the actions.

This occurs, IMHO, when there are apparently conclusive reasons to support inconsistent propositions.

In the case of this puzzle, we have two arguments (both of which seem equally strong) for making opposite choices. The question is whether it succeeds in making the opposing arguments equally strong. If it doesn't, then there actually is no puzzle (or, to put it another way, the puzzle will have been resolved). I don't think the two arguments are in fact equally strong, for, given the conditions of the puzzle, one can find a reasonable explanation for the alien being's ability to predict correctly. And in that case, the argument for taking A is stronger than that for taking both boxes.

If the alien being predicts in the manner of a scientist, that means that there is a certain state of affairs, A, which holds at some point in time prior to your decision and the prediction, and which causes both. This connection between the prediction and the decision is what prevents your actions from being probabilistically independent of the statements concerning the boxes. And it is realizing this that makes it rational to take A only (ie, it is what invalidates the dominance principle).

In order to explain why taking only A is the more reasonable decision, let's first consider what it means to predict something. Prediction can mean at least one of two things. There's scientific prediction, where someone has observed similar conditions many times and predicts the outcome of a situation based on this experience (and on the assumption of some principle of uniformity in nature).

This is how you predict that what goes up must come down, and how the weatherman predicts what tomorrow's weather will be like. And then there's "crystal ball" prediction a la Nostradamus. This second kind of prediction is tantamount to information traveling back in time.

One can certainly argue that both predictions will lead to the same solution to the puzzle, but since my preference is for scientific proof, I will go for taking just A.

Here is where the interaction between the prediction and the event predicted comes in. In the case of this puzzle, there is no causal interaction. Therefore, I will choose A, as the choice of taking both boxes makes it less likely to make a million pounds.

Putting it another way, the risk/reward of taking both boxes is a mere thousand pounds more, which is monetarily/logically unjustifiable (too risky!) when compared to the risk/reward of taking a cool miilion pounds.

[Rob-hin]

If he can really predict the futur, he knows if you will pick A or A & B. No mater what you decide at the last minute.
Indeed he can't change the money that is in it already, but he can't read your mind... only the futur. Only that can tell what you'd do. So he won't need to change what is in the boxes cause he'd already know.
But he could be wrong...

I'd pick only A.