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I believe the Newcomb's paradox is a significant problem in philosophy, since I regard it as a well formulated version of the lazy argument (under certain conditions) with real life implications.

Here is the original problem: There is a predictor [who is never wrong], a player, and two boxes designated A and B. The player is given a choice between taking only box B, or taking both boxes A and B. The player knows the following:

Box A is clear, and always contains a visible $1,000.

Box B is opaque, and its content has already been set by the predictor:

If the predictor has predicted the player will take both boxes A and B, then box B contains nothing.

If the predictor has predicted that the player will take only box B, then box B contains $1,000,000.

The question is, what is the rational choice, taking only B or both A and B.

Now consider this practical variant of the experiment. Instead of the predictor, there is an honest person who promises to fill box B with $1,000,000 if and only if the player chooses only B, and he will do this after the player has made his choice.

I believe in this variant, it's common sense that the rational decision is to choose only B. Now, if we assume that determinism is true, then the content of B is already determined before the player makes his choice. This seems to be equivalent with the original problem. How come choosing only B is not much of common sense in the original problem?

Bonus: I believe the same can be said about the lazy argument. If the predictor has predicted your grade in the tomorrow's exam and has written it on a piece of paper, it seems controversial whether you should study or not. Assumption of determinism also implies your grade is already determined. However, it doesn't make the decision about studying that controversial.

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Newcomb's paradox was discussed at length by philosophers, with the issues of determinism, free will, time travel, etc., brought in. What it turned out to be, however, is an analog of Bertrand paradox which asks for the probability that a "random" chord of a circle is longer than the side of the equilateral triangle inscribed into it. Depending on how "random chord" is interpreted one can derive 1/4, 1/3 and 1/2 as valid answers. This shows that "intuitive" descriptions may be too vague to produce well-defined probabilistic problems because the mechanism for producing the random variable is not specified.

In The Lesson of Newcomb’s Paradox Wolpert and Benford showed that Newcomb's scenario has the same flaw, and that it has it regardless of whether the boxes are filled before or after the choosing. Which means that determinism, free will and time travel are moot.

"We show that Newcomb’s scenario does not fully specify the probabilistic structure underlying the game you and W are playing. The two “conflicting principles of game theory” actually correspond to two different probabilistic structures, i.e., two different games. So there is no conflict of game theory principles in Newcomb’s paradox—simply imprecision in specifying the probabilistic structure of the game you and W are playing. Once that probabilistic structure is fully specified, the game is fully specified. And once the game is fully specified, your optimal choice is perfectly well-defined, and the paradox is resolved.

After establishing this we go on to show that the accuracy of the prediction algorithm in Newcomb’s paradox, the focus of much previous work, is irrelevant. We also show that Newcomb’s paradox is time-reversal invariant; both the paradox and its resolution are unchanged if the algorithm makes its ‘prediction’ after you make your choice rather than before."

The details are technical. There are two specifications of the problem, which they call the "Fearful" and the "Realist". The OP version essentially prescribes the "Fearful" specification:

"Fearful interprets the statement that ‘W designed a perfectly accurate prediction algorithm’ to imply that W has the power to set the conditional distribution P(g|y), to anything it wants (for all such y that P(y)=/=0)... Realist interprets the statements that ‘your choice occurs after W has already made its prediction’ and ‘when you have to make your choice, you do not know what that prediction is’ to mean that you can choose any distribution h(y) and then set P(y|g) to equal h(y) (for all g such that P(g)=/=0). This is how Realist interprets your having ‘free will’."

Whichever interpretation one thinks is the "right one", the assumed "powers" are mathematically inconsistent. In other words, the "paradox" comes down to incoherent interpretation of Newcomb's vague formulation that assumes mutually exclusive things.

  • Thanks a lot. I can't see what the role of probability is in Newcomb's problem. We have bunch of 0 and 100% probabilities, for which we can use the true or false states. For example, the argument for choosing A+B (dominance principle) is based on the assumption "the content of B doesn't change if I change my decision", which is false. – Asmani Apr 15 '18 at 9:26
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    @Asmani It is awkward to phrase statements about conditional probability without probabilistic language even if the probabilities involved are only 0/1. The heart of the issue is that the conditional distributions P(g|y) and P(y|g) can not be independently chosen. Verbal formulations only tend to confuse the issue. Also, it is beneficial to see what happens when certainties are replaced with likelihoods, this is how Nozick originally showed that determinism is irrelevant – Conifold Apr 16 '18 at 20:38
  • I don't have access to the paper. Can you please clarify what P(g|y) means? Probability of what under what condition? – Asmani Apr 17 '18 at 14:20
  • @Asmani They have a freely available version on arxiv What does Newcomb's paradox teach us?, P(g|y) is defined on p.5. – Conifold Apr 17 '18 at 17:56

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