It's high time I posted another paradox blog. This time, it's my favourite - Newcomb's Paradox.
Now before you click that link, first read through the description and decide what you would do in this circumstance and why:
You have been taken by what is clearly a highly advanced intelligence. It obviously has either sophisticated technology or magic, based on the circumstances in which you have found yourself talking to it. It makes an announcement.
"I mean you no harm. I just wish to play a game. In front of you are two boxes, which for your convenience I have labelled A and B. Box A contains £10,000 - an amount I believe you will find significant and tempting, but not quite life-changing.
"Box B may or may not contain £10,000,000 - an amount that quite certainly is life-changing. Whether I have placed that sum in box B is not random. I have used my vast and inestimable powers to study your mind before I brought you here, and I have made a prediction about your behaviour.
"Your two options are to take the contents of both boxes, or to take merely the contents of box B. That is you may take A plus B, or B alone.
"If I believe you will take only box B, I have placed the £10,000,000 in that box. If I believe you will take both boxes, I have left box B empty.
"I have played this game a thousand times before with other people, and I have never been wrong in my prediction.
"If I believe you will try to be awkward and make your decision using, for example, chance, then I will have left box B empty.
"Choose now whether to take just box B, or to take both."
What do you do? Think about it for a moment before reading on.
The problem here is that both options have extremely convincing motivations. If we consider that the being is telling the truth (which it is, or the puzzle may be rather dull), then we might believe that it is wrong less than one in a thousand times, say one in ten thousand (the precise values won't matter). If we look at the relative benefits and likelihoods, multiplying out our expected payoff then:
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- taking both boxes will net us £10,000 with a probability of 0.9999 and £10,010,000 with a probability of 0.0001. A pay off of £11,000 expected.
- taking just box B will net us 0 with a probability of 0.0001 and £10,000,000 with a probability of 0.9999 - a pay off of £9,999,000.
Clearly it is in our favour to take just box B.
Alternatively consider this argument - whatever you do now cannot have an influence back in time. The contents of the boxes are fixed. So just take whatever is there and go on your way happy with whatever you have earnt.
The Wikipedia link goes into some detail of the discussions around this. Suffice to say, I have no entirely satisfactory argument for one over the other.
Which option would you go with, and can you explain why?
So you either believe the alien can predict your actions based on observation, or you don't. If you do - B. It you don't (or you're not sure) take both and leave it to chance (you'll get away with at least 10,000 and a life-changing experience). So it's down to whether you believe the alien, and whether you do or you don't determines the problem you're trying to solve.
I'd choose B. If woke up to be facing and chatting to an alien, would I believe that he could predict my actions (to a certain probability) based on observation (or analysing my mind, which presumable holds a record of my every action)? Yes. And I don't think I understand the problem.. No influence back in time is needed - the alien has a theory/methodology with enormous predictive power (to 1000/1000) - what is wrong with that other than I can't immediately explain that theory? Mmm, maybe I'm missing something.
Of course, you are in a special position to argue that last time i believed in someone's ability to read from behaviour alone I ended up being very very wrong... so yes admittedly I'm a bit of a fool, but then again you're no alien ;)
Personally I think I agree with you. It might appear to strike at free will to allow this, but I see no problem with someone being able to anticipate your actions meaning you don't still have a free choice. I'm quite comfortable with the idea that my actions might be predicted that well (to bring up another example, I don't think it impacts my free will to have someone infer with a high degree of probability that I will never murder someone in the course of my life).
However, it's absolutely clear that by the time you're asked the question, taking both boxes is guaranteed to net you more money than taking just box B, isn't it? So why do we both feel we should not take that option? The logic appears to be watertight that one should take both at that point, but I'm not willing to follow that through.
It's very similar to Kavka's toxin paradox.
I agree, free will is not directly in question here, is it? I thought that was precisely the reason why this sentence
"I have played this game a thousand times before with other people, and I have never been wrong in my prediction"
was there. Suddenly it's not about deterministically knowing someone's actions, just about being very good at it, in a probabilistic way. And that makes it OK.
Yes, it's clear you're guaranteed (with p=1) *some* money if you take both, but if you *believe* the alien, you're guaranteed *a lot of money* (with p=0.999). I don't know how to quantify the probability that the alien is truthful, but I also don't think that is the point here.
I fell asleep thinking about this (damn you), and I've decided that the question itself contains a contradiction. Suppose that we accept with 100% certainty that the Being is never wrong. Then the question "What do you do?" is meaningless. If a friend of yours secretly peeked inside the boxes and whispered the contents to you, this would tell you what you would have done with 100% certainty. Of course you can then choose to do the opposite, but now we've cheated the Being, and that doesn't contradict the fact that the Being would have been right had we not cheated. The point is that the question "What do you do?" itself presumes that the outcome is not determined yet, but it is. This doesn't mean you have no free will, it just means that the Being knows exactly how you will use it, and there is only one possible outcome.
Now consider the case where we know with very high certainty (beyond reasonable doubt) that the Being is not wrong this time. I don't think this changes things very much. Now the question "What do you do?" is just practically meaningless because there is practically only one outcome, just like I practically can't quantum tunnel myself through a wall. If my friend cheats and tells me the box contents, I believe beyond reasonable doubt that I would have done what the Being predicted.
I think the question only makes sense when you reasonably suspect that the Being can be wrong. In that case, we believe that there are several possible outcomes and that our choice will make a difference. Then, since your choice doesn't affect what's in the boxes, I don't see why you wouldn't just take both boxes and hope the Being thought you'd take one.
Newcomb's paradox is frequently framed in terms of a Being with near certain predictive power, but I don't think you even need to exceed reasonable doubt if you are permitted to adjust the sizes of the winnings. I suspect you can still reach a position where the expected payout is larger if you take one box than if you take two, but the dominant strategy remains two boxes?
I've not tried to produce probabilities and prizes to produce this result yet but I expect I'll follow this post up soon!
I think the size of the winnings is irrelevant. I'm saying that we are asking ourselves a nonsensical question so long as we believe the Being is right. It's like saying "I don't believe in Atlantis; should I vacation in Atlantis?" My asking the question contradicts my non-belief in Atlantis, so the question is nonsense. The situation gets only marginally better if I say "Atlantis almost certainly doesn't exist; should I vacation there?" I can't rationally consider even starting on a trip to Atlantis if I'm virtually certain that it doesn't exist. Only when I might reasonably believe in Atlantis is the question internally consistent. Note this is *not* the same as not believing in Atlantis while searching for it just in case I'm wrong.
It doesn't matter whether Atlantis actually exists. If you've been there but I haven't, and I'm somehow certain it doesn't exist, I am being self-inconsistent by even considering vacationing there.
Similarly, if I believe that the Being can predict my actions, and if I know that he already has, that is equivalent to saying that I believe there is but one outcome: I will do what the Being predicted. Asking myself the question "What should I do?" presumes that there are still outcomes open to me, which contradicts my belief that there is only one possible outcome. Only when I suspect the Being could be wrong does the question become rational combined with my beliefs.
Apparently I need to do some reading on causal decision theory.
Chaz,
You've gone full circle - you're back on the B-side.
If you believe the Alien, then there's only a paradox IF you have a problem with the consequences towards free-will. If you believe the Alien, there's only one outcome and you come away with 10M - done! No paradox, just lots of money. If you DON'T, then take A+B and be happy with what you get - no paradox, just a guaranteed sum of money.
I refer back to my initial comment: "So it's down to whether you believe the alien, and whether you do or you don't determines the problem you're trying to solve."
The "paradox" (and tomorrow we can discuss whether such a word is even meaningful) I believe only occurs if you think this somehow compromises free-will beyond what is comfortable. Personally, I don't think it does. Perhaps that is why I have no problem in believing the alien, in taking box B and partying like I just won £10,000,000.