Newcomb's Problem

[pigswill]

I suspect that this problem is very little to do with logic or quantum mechanics or free will but individual psychology.
You can either accept the idea that your behaviour is predictable or you can reject the idea.
For instance. A person drinks coffee with sugar. The last hundred cups of coffee they drank had sugar in them. Every time they made a coffee they had a choice, an exercise of free will, about whether they added sugar, and every time they did indeed add sugar.
There's no guarantee that the next cup of coffee they have will contain sugar. Maybe they'll have ran out of sugar, maybe someone's put salt in the sugar-bowl, maybe their coffee gets spilt by Shroedinger's (sic) cat. But its very likely that they will indeed have sugar in their next cup of coffee.
You may not like the idea that your behaviour can be predicted with a very high degree of certainty, but that doesn't change the fact.
If you accept the idea that you're predictable you get $1,000,000. If you reject the idea you get $1000.

 

[Kennigit]

If you accept the idea that you're predictable you get $1,000,000. If you reject the idea you get $1000.

Not really. You only get $1,000,000 if Omega puts money in box B. Your choice has no relevance to what is actually in box B. If I choose box A and only get 1,000 dollars, then congrats to Omega for correctly predicting; but in my situation there was only $1,000 available, meaning I would have gotten $0 if I choose box B.

On another note about statistics of this situation: There isn't any "likelihood" that box B has a million dollars or not, and none of this "If Omega is able to predict 50.05% it is better to take box B". When you are presented with the choice of either both boxes or one box, the money is already there and will not change (as described in the OP). If you for some reason calculate that there is a 65.42345% chance that box B has a million dollars, it doesn't make a difference what you choose.

Also, about relying on past trends, you can think back again to coin flips. If I flipped a coin 100 times, and got 60 heads and 40 tails, does that make the chance of my next flip be 60% for heads and 40% for tails? No. The past flips have no relevance to the current flip.

 

[El_Machinae]

"He gives millions to people who're latently faithful, and $1000 to those who are doubters"

 

[Darth_Pugwash]

Well, there is a system like that which Omega could be working by, which is a bit too large to rightly be called "fudging". He could be using quantum suicide, as I mentioned earlier. For those unfamiliar with this, it works sort of like this:
1) Flip quantum coin to determine whether to predict that person will pick box B or not
2) Use many-worlds interpretation to create one world with and one world without money in box B
3) Present boxes and game to person
4) If person picks wrongly, destroy that universe, leaving behind only the universe in which the person picked according to the prediction.

Hence, the fact that our universe exists, by a variant of the anthropic principle, shows that it's the one in which everyone picked the box that Omega "predicted".

I've read about quantum suicide before while article hopping on Wikipedia, it's kinda a fun idea in a completely counter-intuitive sort of way.

So, if Omega is using quantum suicide, the possibilities run like so:

Pick both:
Result 1: $1,000
Result 2: Wiped from existence

Pick B:
Result 1: $1,000,000
Result 2: Wiped from existence

I have a 50% chance of being wiped from existence, no matter what I pick. Ouch. Of the two results where I survive, box B gets me more money. I have no idea which is more likely. So I pick B, cross my fingers, and hope that the universe does not fall apart around me!

Not really. You only get $1,000,000 if Omega puts money in box B. Your choice has no relevance to what is actually in box B. If I choose box A and only get 1,000 dollars, then congrats to Omega for correctly predicting; but in my situation there was only $1,000 available, meaning I would have gotten $0 if I choose box B.

Again, it is not disputed that the contents of the boxes remain constant from the moment they are created. Here's the rub: what if Omega knows which option you will take, when he creates the boxes! Say he's a time traveller. Say he's a mind reader. Whatever, it doesn't matter. The upshot is that if you pick both, then box B will always have had nothing in it, and if you pick B, then box B will always have had $1,000,000 in it! Becuase Omega will have predicted your choice like he was 'predicting' the colour of the sky!

On another note about statistics of this situation: There isn't any "likelihood" that box B has a million dollars or not, and none of this "If Omega is able to predict 50.05% it is better to take box B". When you are presented with the choice of either both boxes or one box, the money is already there and will not change (as described in the OP). If you for some reason calculate that there is a 65.42345% chance that box B has a million dollars, it doesn't make a difference what you choose.

Imagine that you, for whatever reason, have to bet money on a football game. You look up some pundits and their opinions on the game:

Pundit 1 has predicted 20/100 games correctly, and thinks that team A will win.

Pundit 2 has predicted 80/100 games correctly, and thinks that team B will win

Would you bet on team A or team B? Whichever team you bet on, you will not influence the outcome of the game, and neither will the pundits. All any of you are doing is predicting! But you can endeavor to match your predictions as closely as possible to observed patterns.

Also, about relying on past trends, you can think back again to coin flips. If I flipped a coin 100 times, and got 60 heads and 40 tails, does that make the chance of my next flip be 60% for heads and 40% for tails? No. The past flips have no relevance to the current flip.

Okay, so Omega has gotten 100/100 predictions correct. We do not know what the 'real' chance he will be correct is. Given this, is it wiser to infer:

A- He has no powers of prediction and has thus far been lucky.
B- He has very good powers of prediction.

The one-boxer says that it is wiser to infer B, however the two-boxer goes against the observed evidence and says that it is wiser to infer A.

 

[Ayatollah So]

He could also have visited a hundred gazillion planets, playing the same game, and it just so happens our planet is the one where he was right 100 times in a row.

Nice one. There's a well-known con based on that theme. You offer free predictions of game outcomes to lots of bettors. After 5 or so, you start charging money for more predictions, to those with whom you've been mostly correct.

My own cheating hypothesis: the alien installs a device into every box B, which contains a million bucks, and which can detect whether box A has been taken, and which vaporizes itself (and the $) when that happens. When only B is taken, the device discharges the $ and then vaporizes itself.

 

[BCampbell]

I only skimmed all of the posts in this thread, but I don't think anyone mentioned this yet.

The original problem states:

Omega has been correct on each of 100 observed occasions so far - everyone who took both boxes has found box B empty and received only a thousand dollars; everyone who took only box B has found B containing a million dollars.

It doesn't say how many people Omega has observed in each case. For all we know, all 100 could have picked both boxes, or all 100 could have picked box B. This is important for everyone arguing over the accuracy of the prediction, because I believe it makes this bit of information from the problem completely irrelevant.

eta: the "solution" seems pretty simple to me after drawing a quick diagram, but maybe I'm oversimplifying. I'd rather let other people come up with the diagram on their own than spoil just yet, though -- or again, maybe it's already been done and I skimmed over that.

oh, what the eff, I'll just put it in a spoiler in case anyone's curious.

Spoiler :

I found it easiest to break this down into the four different possibilities:

Decision | Prediction | Result
A&B.......A&B..........$1000
A&B.......B only.......$1,001,000
B only....A&B..........$0
B only....B only.......$1,000,000

Once it's broken down like this, I see that regardless of the prediction, I will always get more money by picking both boxes. In addition, picking box B is the only way I could walk away with nothing. Since the money is already either there or not, I'm not totally sure I understand any reason not to take both boxes.

Put another way, box B either has $0 or $1,000,000. I will get the amount in that box regardless of my decision, so I might as well take both and get the extra $1000.

I think I can understand the B only position though, as it relies on the predictive ability of Omega. Thus it seems like picking A&B by default would cause B to contain $0; Omega surely knew you'd pick both. However I think breaking it down like above defeats this, as it clearly shows that, no matter what's already in box B, taking both gives you $1,000 more. That is, if Omega predicted you'd take box B, and it thus contains $1,000,000, you might as well take both boxes anyway and get the extra $1000.

I think it's also worth noting that picking both boxes is the only way to get the maximum amount of money.

Anyway, for the reasons above, I would pick both. But again, maybe I'm oversimplifying.

 

[Gorrable]

This is literally the first time I've ever ventured into Off Topic, but I'm liking this thread - handed in my dissertation about a week ago. On Newcomb's Paradox.

Am pretty knackered now, so I don't have the energy to read through the whole thread, and skimming makes it fairly clear that a lot of the arguments (that aren't based in technical theories of logic and rationality) have been covered already. I will share a funny story I included in my dissertation, which, as you'll see, may well loose me some marks.

"Whenever I'm walking home drunk, I tend not to listen to my Ipod - mostly because I need to direct all my energy towards not falling over. Interestingly, this tends to correspond pretty well with my hangovers - if I'm too drunk to listen to music, that's a pretty good indication that I'm going to get a hangover. The lack of music doesn't cause the hangover, of course - that would be the many £1.40 pints at the SU.

The thing is, I was walking home last night, and I thought - wait a minute, I could listen to my Ipod now. I don't really want to - I'm trying to remain upright - but if I do, then there's a good probability I won't get a hangover. This is because we calculate probability simply by relative frequencies - 99 times out of 100, if I listen to my music, I don't get a hangover. Should I listen to music?

Well, obviously, no. I will still get a hangover, whatever I do then - the die has already been cast. To use the language of the first post - my expectation of listening to music is high, in fact, I could perform the exact same calculation as the first post does to prove that my expectation is higher for listening to music than not. None the less, I'm going to get a hangover anyway. If I don't want to listen to music, I shouldn't."

That's not particularly well explained - the idea is, what matters isn't the basic probability, it's the probability that your action will cause the outcome. Music doesn't cause hangover, beer does, and I've already drunk the beer. Choice doesn't cause prediction, the predictor does, and he's already made his prediction.

Although I do like the idea about the boobytraps.

 

[raketooy]

If such a superintelligence came down here, I could well believe that it is somehow, I need and may not know how, but somehow in its hands to ensure that its prediction will hold true. So I would take one box. One thousand is not that big of a loss, anyway.

 

[BasketCase]

Isaac Asimov had this opinion of the problem:

I would, without hesitation, take both boxes . . . I am myself a determinist, but it is perfectly clear to me that any human being worthy of being considered a human being (including most certainly myself) would prefer free will, if such a thing could exist. . . Now, then, suppose you take both boxes and it turns out (as it almost certainly will) that God has foreseen this and placed nothing in the second box. You will then, at least, have expressed your willingness to gamble on his nonomniscience and on your own free will and will have willingly given up a million dollars for the sake of that willingness--itself a snap of the finger in the face of the Almighty and a vote, however futile, for free will. . . And, of course, if God has muffed and left a million dollars in the box, then not only will you have gained that million, but far more imponant you will have demonstrated God's nonomniscience.

The idea of an omniscient Being who already knows your future seems to really rub some people the wrong way.

Personally, I prefer that the paradox be described with a million dollars in both boxes, because as Rake put it, a thousand dollars is not a lot when you consider what might be in the other box.

 

[Mise]

Choice doesn't cause prediction, the predictor does, and he's already made his prediction.

Clearly, listening to music doesn't cause hangovers. But it's less clear as to whether making a choice in the future causes the predictor to predict something in the past.

Ayatollah So called it .... something or other, a "determinative relation"? And I believe that this is logically valid -- the order in which cause and effect occur bear no effect on rational decision making.

If I put two boxes in front of someone, both transparent, one of them containing $1,000, the other completely empty, and I told them to take the contents of one, both, or neither of the boxes, I could predict with 100% success rate that they'd pick the one with $1,000 in it.

That works because everyone accepts that if a box contains $1,000, and you take the contents of that box, you will be $1,000 better off; and if a box is empty, and take the contents of that box, you will be no better off.

Newcomb's problem would be exactly the same trivial problem, if everyone accepted that effect can precede cause.

 

[BasketCase]

How about this version:

The first contains a million bucks. The other contains either FIVE million....or nothing. You only get to pick one box. Which one?

I've always been puzzled by the way the paradox was originally phrased.

 

[Mise]

How about this version:

The first contains a million bucks. The other contains either FIVE million....or nothing. You only get to pick one box. Which one?

I've always been puzzled by the way the paradox was originally phrased.

That's a completely different scenario, because it doesn't take Omega (or God or whoever you want to call it)'s prediction, and allocation of money based on that prediction, into account.

And the result is simply how risk averse you are...

 

[BasketCase]

The rest of the paradox remains unchanged. The Omega fills Box Number Two the same way: if he thinks you're gonna take Box 2, he left it empty. Otherwise he put five million in it.

 

[Mise]

The rest of the paradox remains unchanged. The Omega fills Box Number Two the same way: if he thinks you're gonna take Box 2, he left it empty. Otherwise he put five million in it.

Knowing that changes the way you play (well, at least I think it does, and that's what makes it self-fulfilling, in my eyes).

Also, you don't only get to pick one box... in the original problem, you could pick both boxes if you want.

 

[BasketCase]

The basic idea behind the paradox is the idea of causality.

The Omega already made the decision and left. He either put five million dollars in the second box, or he did not......or did he? Can your choice somehow influence whether the second box is empty?

 

[Ayatollah So]

The thing is, I was walking home last night, and I thought - wait a minute, I could listen to my Ipod now. I don't really want to - I'm trying to remain upright - but if I do, then there's a good probability I won't get a hangover. This is because we calculate probability simply by relative frequencies - 99 times out of 100, if I listen to my music, I don't get a hangover. Should I listen to music?

That isn't how I calculate probabilities. I use all the evidence at hand and my best theories. If I were in that position, I'd note that I don't feel like listening to music, therefore I have a 99% chance of a hangover, no matter what I do.

 

[WillJ]

Short version: "determinative relation" just means cause & effect.

Longer version: only, not exactly. See, the words "cause" and "effect" imply time-order. If two events are linked by a law of nature, we call the first one "cause" and the second one "effect". Now, there may be more to the average Joe's conception of cause and effect than just time-order plus laws of nature. But the rest is dispensable, I think.

So what is a determinative relation? Suppose you have an event, X, at a certain time, and you have some laws of nature (conservation of energy, of momentum, etc. etc.). Suppose that from these facts you can logically derive another fact W (Y) earlier (later) in time than X. Then X is determinative of W (Y).

For rational decision making, time-order doesn't matter if your choice is determinative of a result.

It doesn't matter for rational decision making, but it matters for consulting. As we consider Newcomb's Problem, we can think of ourselves as consultants to an imaginary person contacted by Omega, advising him on whether to choose B or A&B. If there is no "determinative relation" at all between Omega's prediction and the person's choice, obviously we should consult the person to choose A&B. If there is a determinative relation, there are two possibilities: either the person is somehow "fated" to choose what Omega predicts (either due to the super-duper-ness of Omega, or some force acting on both the person and Omega), or with the help of some weird physics, the person's choice is actually "in control of" Omega's prediction, despite it having happened in the past. I'm pretty sure these are conceptually distinct. The distinction is irrelevant for rational decision making; in either case, the rational choice is B. But it matters to us as consultants; in the latter case, we should argue to the person in favor of B, while in the former case, it's useless consulting with him, as regardless of what we say, he'll choose whatever he was fated to choose, with no impact from us. (Well, we might have an impact, if the person was fated to choose a certain way because of us, but that'd only be the case if we were fated to advise a certain way, in which case imagine a person consulting us on how to consult ... possibly ad infinitum, with the idea of "consulting" being my way of expressing an escape from determinism).

Now, if we imagine ourselves as the person, and as our own consultant, the same applies: our deliberation over Newcomb's Paradox is only relevant if there's no determinative relation or if we're in control of Omega's prediction. If we're fated, then, well, we're fated, and there's no sense in deliberating (but of course, in that case we're probably fated to deliberate!).

...This is one of those cases where I haven't fully worked out where I'm going, or if I've already gotten to where I want to be, or how my location relates to others' locations, or even if I started from the right spot, so I'll stop typing for now....

 

[Phlegmak]

Erik Mesoy, please provide some explanation. How is this a difficult choice?

 

[Ayatollah So]

If there is a determinative relation, there are two possibilities: either the person is somehow "fated" to choose what Omega predicts (either due to the super-duper-ness of Omega, or some force acting on both the person and Omega), or with the help of some weird physics, the person's choice is actually "in control of" Omega's prediction, despite it having happened in the past.

What defines "fated"? Does it mean by definition, a course of action/events that will happen regardless of what reasons and arguments are brought against it? In that case I don't think Omega's success record gives us any reason to think the person's decision is fated.

About "weird" physics: there's nothing weird about physics that has determinative relations extending both forward and backward in time. Newtonian physics is like this. For example, from the total momentum of a colsed system today, you can equally well derive the total momentum of the system yesterday, or tomorrow.

 

[Erik Mesoy]

Erik Mesoy, please provide some explanation. How is this a difficult choice?

It depends on how you want to break the situation down.

You can do it like this:
*Either I pick box B, which Omega has probably predicted, seeing as he's been predicting well so far, and thus I get $1M.
*Or I pick both boxes, which Omega has probably predicted, seeing as he's been predicting well so far, and thus I get $1000.
*I should pick box B.

Or you can do it like this:
*Either Omega has predicted, correctly or not, that I'll take box B, in which case taking both boxes earns me $1M+$1000 and taking box B earns me $1M.
*Or Omega has predicted, correctly or not, that I'll take both boxes, in which case taking both boxes earns me $1000 and taking box B earns me $1M.
*I should pick both boxes.

I'm not sure it's a "difficult" choice. I think the answer is obvious and it appears that you do so to. It most definitely is a controversial choice, though, as suggested by the number of people disagreeing and talking past each other for why their choice is correct.