Newcombe's Paradox?

The recent post about another paradox reminded me of something I came across some years ago in a column by Martin Gardiner in Scientific American. It may have been called “Newcombe’s Paradox”. At the time, there was no ‘resolution’ to it (not that such a thing is possible in a true paradox). I’m curious to know if there’s been any developments or if any of the Dopers would care to offer their insights.

The problem is this:

**Suppose there is a very clever, very insightful being who is an expert on predicting human behaviour. In fact, this being has been nearly 100% accurate in predicting the behaviour of individual humans since records were first kept, eons ago. Thankfully, the being is benevolent and wants to reward humans, although he doen’t like greediness. He offers you, a human, the following scenario:

There are two closed boxes “A” and “B”. The being gives you a choice. You can take both boxes or only box “B”.

Now, in box “A” the being places $1000.

In box “B” he places either $0 or $1,000,000 depending on whether he has predicted that you will take both boxes (you’re greedy) or just box “B” respectively (not greedy).

You make your decision after he places the cash.**

What would you choose in order to maximize your gain?

The main thing I recall about this paradox was that whichever “side” people take, they are convinced that those taking the other viewpoint are being either silly, argumentative or just plain stupid!

(BTW, I apologize if this “paradox” is now passe or hackneyed. If so, I missed it.)

I’ve always heard it that the being is truly omnipotent, and that he does predict your action accurately, no matter what. When stated like this, it’s just the same as any number of time-travel paradoxes. What you have is a causal relationship (which box you will choose determines the placement of cash) happening in reverse. This process defies the laws of Physics as we know them, so it’s sort of like saying, “What if something happens and it doesn’t happen at the same time?”

However, your statement of the paradox involves a being which could, in the realm of possibility, be false. Your job is to see that he’s false. All you have to do is figure out which outcome he predicted, and go with the opposite. To do that, you have to have more information - how did he determine which box you’d probably take? It kind of reminds me of Vizzini’s death scene in The Princess Bride, which, if you remember, shows you that this is not something to be taken lightly!

I don’t know “the answer,” but I’d go ahead and take both boxes.

Why? Well, under the circumstances, I AM greedy! The whole point of this exercise is to get as much cash as possible, and I wouldn’t be taking part in this exercise at ALL if I weren’t determined to make a profit, now would I?

Now then, if this genius is REALLY as good at reading people as he’s supposed to be, he’ll have figured out in an instant that I’m a greedy slob. So, in all likelihood, he’ll have put zero in box B. So, I might as well take both boxes. That way, I get a thousand dollars for sure, plus the TINY possibility of winning another million (just in case the “genius” might have screwed up).

Paradoxes rest on the initial assumptions. The existance of a paradox means you should examine the assumptions to seek resolution. The version of this paradox that I heard also specified that the being has never been wrong. I think the existance of such a being is questionable.

I recently read a discussion of this paradox – didn’t say whether it had been resolved, but it did mention one thing.

NOTE : If you haven’t thought about which answer is correct, do so now before you read on unless you really don’t care.

Sounds like an annoying e-mail forward, doesn’t it?

Anyway, the observed psychological(?) reaction (supposedly) is that those who believe in free will will take both boxes. Those that don’t, i.e. believe in determinism will only take the one. There didn’t seem to be a clear reason why, other than that it seems obviously inferable.

I chose the two boxes, by the way. At first I thought it was greed, but I suppose one could argue greed points to the other option as well. And I don’t think anyone crazy who only takes one box. (Actually, I only took box A and left box B sitting there. You can all have the money in it if you want.)

panama jack

“Cast but a glance at riches, and they are gone” - Proverbs 23:5

Not one person in the entire race over all the eons ever told this hemi-demi-semi-potent excuse for a prognosticator to shove either one, or both of his boxes up his ass?

Put me down for a pass.


<Drive by URLing>

The Newcomb Paradox

This site has a very good analysis of the problem, and a little narrative that will make it easier to believe the interragator really can predict what you’re thinking.

Panamajack – your evaluation depends on the assumption that determinism is not compatible with free will (which it is, silly). Still, the problem does relate to determinism, obviously.

I believe that the being in the problem is not supposed to be identified as omnipotent, merely that in playing his little box game he has been correct every time so far out of countless tries.

I’m with DrMatrix. Show me this being who predicts everyone’s actions accurately and (if he tests out) then we’ll have a paradox. Otherwise we’re just playing games with words.

But assuming you come up with such a person, what happens if I flip a coin to decide which box(es) to take? Can this guy predict the fall of the coin? If I take only B and it’s empty, can I sue him for a million bucks?

One of my favorite books, Labyrinths of Reason by William Poundstone (quick amazon description here) discusses this paradox in detail. Poundstone’s style is to discuss one side of the paradox to convince you, and then switch sides and argue the other way. Since it’s a paradox, he goes a few cycles around. I enjoyed the discussion, so it’s recommended. Unfortunately, Amazon seems to be out, but I imagine you can find it somewhere else.

I’d take both boxes. Its not the money I just like boxes!

Put me down for box B only. Given his track record (100% correct since record-keeping began), the chances that he’ll screw up on my turn are vanishingly small. Thus, the expected outcome of only taking box B is something like E=$999,999.9999999999

It’s as close to a sure thing as you can get.

You see, what this paradox really illustrates is the two different ways people attack this problem: either you want to get everything possable (take both boxes), or you trust that the being knows his stuff (box B only). Both trains of though make perfect sense to the person who thinks that way, while the other way just seems foolish.

Thanks for the comments and input. I had missed your link, Hunsecker - probably because I spelled it wrong.

I agree with those who feel that the existence of the ‘paradox’ should be taken as indication that such a being cannot exist.

Hold on here… The problem as stated is trivial… If this being (let’s call him Bob) is actually guaranteed to get it right, then it’s just the same as if he waited until after you told him your choice. If you say he’s guaranteed, and still say that you’d trick him, then you’re changing the initial assumptions.

The real paradox comes in when, instead of him basing his decision on how many boxes you choose, he bases it on how greedy you are. If I know that he’ll give me the million if I choose just one box, then I’ll choose one box. However, the reason I’m doing this is because I want the million bucks, so I’m greedy. Of course, if he knows I’m greedy, he won’t give me the big money… but if I know that he won’t give me the big money, and I choose it anyway, then I’m not greedy…

In this case, I’d say that the best solution is to just ask Bob which box I should choose.

[hijack]Oh, and by the way, in the iocane duel in The Princess Bride, Vizzini wasn’t basing his reasoning on the levels-of-deception thing; he was trying to read the Man in Black’s reactions to his proposals for the drinks. If MiB gets nervous when V suggests that they each drink from their own cup, then the poison is in MiB’s cup. Of course, there’s still bluffing involved, in that (were the contest fair) MiB would be trying to make V think he’s nervous when he’s not, and vice versa.[/hijack]

Okay, the discussion and links given here make it pretty interesting, but I remain with a lack of understanding on the related issue.

Namely, how is it possible for one to believe in free will and determinism? VarlosZ, you’ve pointed this out but I didn’t know it then, and I don’t understand it yet. Could you explain it? The determinism as I interpreted it means you believe all actions, yours included, are decided already, you can’t make a choice even if you appear to. Free will allows a choice; this is very brief but is my brand of determinism (or free will) wrong?

Hoping to fight ignorance (and attend N.I.T. someday).

panama jack