I was briefly told about Newcomb’s paradox the other day. I’m not sure I heard it correctly or not, but as I understand it, it goes something along the lines of:
A psychic has two boxes, one with 1000 dollars in it and another that could be empty or filled with 1 million dollars. It’s contents depend on what box you choose. The psychic says that if I pick the empty box only, then it will have 1 million dollars in it, if I take both boxes, then I only get 1000. The psychic (who is never wrong) knows beforehand which box I will pick, so the money (or lack thereof) is already in the box.
Okay, I don’t understand how this is really a paradox, can someone explain it to me, because the answer seems blatantly obvious.
You choose the empty box, that will then have 1 million dollars in it, because you were predicted to take it.
Well, think about it this way: The psychic has already made the prediction, and the box already has either $0 or $1000000 in it. So if you pick both boxes, then you’re $1000 richer than if you’d picked only the one, right?
Cribbing from another explanation I read somewhere: suppose this was a gameshow, “Newcomb’s Paradox”, and you were a contestant. The boxes are placed on stage, and each have one glass side, so that the audience can see in. You’re led out on stage (on the other side of the boxes, so you can’t see in), and you get to make your choice.
Suppose you choose only one box. If it’s empty, the audience thinks “boy, what a maroon. He coulda had $1000 if he’d a picked both.” If it contains a million, he audience thinks “boy, what a maroon. He coulda had another 1000 smackers if he’d a picked both.” So why wouldn’t you pick both?
Note that I can argue the other side, too. That’s what makes it a paradox.
I took a class on paradoxes once, and the professor argued that there’s no rational reason to choose either way. Me, I’d take both–$1000 is quite a gift.
Actually I think that would change it a bit…The way I’m thinking it, the psychic’s always right, so therefore the psychic would know whichever one you picked.
There is no paradox if the psychic is 100% correct, and this fact is known. The situation is self-consistent.
If you choose both boxes, this fact was known by the psychic, and you’ll get 1000.
If you choose the ‘empty box’ (closed box) this fact was also known, and you’ll get 1000000.
It is impossible to choose the closed box and get nothing. It is likewise impossible to choose the open box and get 1001000. Neither of these states exist.
However, the instant the psychic becomes probabilistic as to the truth and you know exactly what that probability is, you can work out the expected value of the outcomes and choose the one that gives you the most money.
If you don’t know the probability the psychic is correct… well, now that’s interesting. I wonder if you could compute the expected value over all probabilities the psychic is correct on both choices…
I can’t see the paradox either.
I totally agree the money’s either there or it isn’t, but the puzzle as stated automatically sets up ‘states’ with certain probabilities.
Trying to argue about the audiences doesn’t really prove anything. It’s like saying that a five card poker hand you can’t see is nonrandom when you can’t see any of the other cards even if someone else can see the poker hand. It’s still random to you. (I am assuming the audience isn’t conveying information to you of course).
The paradox lies in “The psychic says that if I pick the empty box only, then it will have 1 million dollars in it” and if you pick both, the box will be empty.
What’s in the second box depends on whether or not you pick the first one, so until you decide whether or not to take the first box, the second box is undefined.
See, if this were a normal, random situation with no foreknowledge, you’d always do best to pick both boxes, because if you do, you’ll either win $1,001,000 or $1000.if you pick both, while if you win either nothing or $1,000,000 if you pick the second box.
However, in this case, with the knowledge you have, you’re better off only picking the second box, because you’ll win the million, while if you picked both, you’re getting $1000.
I disagree; there’s still a paradox at work. Let’s go one step farther: suppose, again, that this was a gameshow, “Newcomb’s Paradox”, and you were a contestant. This time, though, the two boxes are clear glass on all four sides, so you can clearly see what, if anything, they contain. Would you still pick only the one box? If so, why? If not, what’s different?
Clearly, to me, you’d choose both boxes in this case. I mean, the money’s right in front of you! But how is this any different from the case where you can’t see into the box? In neither case will choosing two boxes make the money disappear, so why not choose both? You’re automatically $1000 richer.
And your analogy to a card game is flawed. When you’re betting based on the unknown, but non-random hands held by the other player, your decisions will increase or decrease the payoff (i.e., you win or lose), in some way that you don’t really know. With Newcombe’s paradox, your decision will not affect whether or not the pile of money is there, so how can choosing only one box do anything but reduce the payoff?
Surely, the audience would see either an empty box, or, a $1m.
If it’s empty they will say to themselves, “What a dozy git this contestent is, he is #going# to choose both boxes.” etc.
(appypollyloggies if that’s what you meant by arguing the other side).
Yes, and, no. Decision and Value Theory (DVT) as spoken about in the link given by dylan_73 says that the contents of two boxes is worth at least as much as the contents of one of those boxes, so you might as well take both.
This opinion is based on the assumption that the payoff-matrix looks like this:
$1m is there $1m is not there
choose one $1000000 $1000
choose both $1001000 $1000
The problem here is that it disregards the way the “paradox” (I see no paradox) is set up to create feedback between decision and outcome, so the matrix really looks like this:
$1m is there $1m is not there
choose one $1000000 infeasible
choose both infeasible $1000
As far as I’m aware DVT has nothing to say about such situations, but common-sense tells us to choose one.
If it looked like this:
$psychic is right $psychic is wrong
choose one $1000000 0
choose both $1000 $1001000
and we have NO probability distribution for the accuracy of the psychic then DVT is simply not going to help us, except to wave its arms around about Maximax/Minimax/Laplacean/Minimax-Regret/a-gazillion-other-decision-criteria, before finally coming clean and saying “Oh I don’t know, why not toss a coin?”