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Imagine there are two boxes, one transparent and the other opaque. In the transparent box lies a thousand units of money. My options are to select the contents of the opaque box only or both the transparent and opaque box. A "predictor" can reliably determine what I shall do before I do it, so if I select the former option, I can expect to obtain a million dollar cheque in the opaque box. However, if I select both boxes, I shall find nothing in the opaque box and a paltry thousand in the transparent box.

As a rational human being, I would simply take the contents of the opaque box and walk away with a cool million. Yet, as I walk away, I notice there is a thousand in the transparent box, so why can't I just walk back and take it as well for a cool million plus a thousand? After all, the predictor already fixed the contents of the boxes before my decision, so what choice I make doesn't matter regarding the contents of each box, so it's best to take both.

In other words, what is the conflict between the principle of dominance and the principle of maximizing expected utility?

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    I can never get myself to understand or believe in the predictor. I don't think this paradox makes sense because of that. Clearly others disagree.
    – user4894
    Jun 19, 2014 at 7:36
  • @user4894: then you are rejecting the premise of the question, and presumably don't find it worthwhile to engage the question. If you do engage the question, however, then you should carefully consider what follows from the premises.
    – Niel de Beaudrap
    Jun 19, 2014 at 9:28
  • One can argue that it only makes sense to accept the dominance principle as a valid principle of rationality when the players' strategies are causally independent of each other.
    – user3164
    Jun 19, 2014 at 9:53
  • @NieldeBeaudrap It's worse than my rejecting the premises. I observe that the premises are inconsistent. If a predictor can exist, what does it mean to say the player is "rational?" The implication is that the player has free will. But clearly the existence of a predictor negates that assumption. If a predictor exists, then the player has no choice and his "rationality" is irrelevant. The problem as stated makes no sense.
    – user4894
    Jun 19, 2014 at 16:33
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    Anyway: it's a logic puzzle. If you insist that the predictor make a factual prediction about your choice as set out in the puzzle, you ought also to assume that any basis whatsoever on which you might make a choice, including apparently random processes, are in fact predictable by the predictor — as set out by the puzzle. If you reject this premise, you are not proposing an answer, you are only rejecting the paradox as not being a question worth entertaining.
    – Niel de Beaudrap
    Jun 19, 2014 at 18:51

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Normal arguments regarding maximising expected utility, presuppose that the distribution of conditions is independent of your strategy.

The counterintuitive (and many would say: counterfactual) element of Newcomb's paradox is that the distribution is said to depend on your strategy. You can understand the game easily enough for two strategies:

  1. Suppose the predictor actually has the described power, and you both believe it does and trust that it will provide the $1M if you intend to take the contents of only the one box. Then, if only for selfish reasons, you'd be a fool to plan to take the second box. And after the reveal, it doesn't matter that you could then grab the second box as well: by hypothesis, you are in that circumstance because of a prediction, assumed accurate, that you won't choose to.

  2. If the predictor actually has the described power, but you either don't believe it does or don't trust it to provide the $1M, you will be receptive to taking both boxes. The predictor, knowing this (by hypothesis) and also knowing whether you would choose to take the second box given the opportunity, will leave the opaque box empty if so. It would in effect be producing a self-fulfilling prophecy, but only because it knows enough about your intentions to do so, and because it has for some reason resolved only to give you $1M under specific circumstances.

Your strategy, in the game, would obviously depend on whether you think such a predictor is possible, and honest. If you don't, you should take both boxes; if you do believe in the predictor, you shouldn't.

If you believe in the predictor but get an empty box, that's tough: you were wrong either about it's ability to predict (if you go away empty-handed) or its intent to act in good faith (if you intended to take only one box but take both after re-evaluating your relationship with the predictor). However, the premise of the puzzle disallows this contingency. You may not believe that the game can be realised in fact, but the premise of the question is that the predictor is both perfect at prediction and honest. And in this case, any doubt of the player is a matter only of not knowing, or accepting, that premise.

If you accept the premise of the question, it's obvious how you ought to act. The only question, then, is whether you accept the premise of the question! If you don't, then it's not that Newcomb's paradox is unsolvable: it's that you refuse to regard it as worthy of serious consideration. And that's bound up with issues of philosophical naturalism (computational complexity in particular), and psychology.

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