Consider The Following

We’re having a busy weekend — among other things, our daughter just flew in from China to stay with us for the Christmas week — and so I haven’t had the time to sit at the computer brooding and writing.

For tonight, then, a logical curiosity you may not be familiar with: Newcomb’s paradox.

Here’s the summary, from Wikipedia:

A person is playing a game operated by the Predictor, an entity somehow presented as being exceptionally skilled at predicting people’s actions. The exact nature of the Predictor varies between retellings of the paradox. Some assume that the character always has a reputation for being completely infallible and incapable of error; others assume that the predictor has a very low error rate. The Predictor can be presented as a psychic, as a superintelligent alien, as a deity, as a brain-scanning computer, etc. However, the original discussion by [philosopher Robert] Nozick says only

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that the Predictor’s predictions are “almost certainly” correct, and also specifies that “what you actually decide to do is not part of the explanation of why he made the prediction he made”. With this original version of the problem, some of the discussion below is inapplicable.

The player of the game is presented with two boxes, one transparent (labeled A) and the other opaque (labeled B). The player is permitted to take the contents of both boxes, or just the opaque box B. Box A contains a visible $1,000. The contents of box B, however, are determined as follows: At some point before the start of the game, the Predictor makes a prediction as to whether the player of the game will take just box B, or both boxes. If the Predictor predicts that both boxes will be taken, then box B will contain nothing. If the Predictor predicts that only box B will be taken, then box B will contain $1,000,000.

Nozick also stipulates that if the Predictor predicts that the player will choose randomly, then box B will contain nothing.

By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents of box B, the fact that its contents are based on the Predictor’s prediction, and knowledge of the Predictor’s infallibility. The only information withheld from the player is what prediction the Predictor made, and thus what the contents of box B are.

So, dear Reader: what will you do? Before you just go ahead and grab both boxes, remember that the Predictor is, effectively, never wrong.

There’s more discussion at the Wikipedia article; you can read it here.

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  1. Kevin Kim says

    Wait—is there a downside to this? If the player has the option of taking either Box B or both boxes, and if Box B contains a million dollars if the Predictor predicts the player will pick Box B alone, then it seems, effectively speaking, as if the player can’t lose: he gets at least a thousand dollars. I realize I’m mangling cause and effect, here, but the problem could be set up this way, assuming the Predictor’s infallibility:

    1. I choose to take both boxes. I gain a thousand dollars.

    2. I choose to take only Box B. I gain a million dollars.

    3. I choose randomly, maybe by flipping a coin, mindlessly opting for either (1) or (2) above. I gain a thousand dollars.

    Messing up the scenario is the fact that you’ve explained how the Predictor works, so I already know what’s going to happen. It’s impossible for me, now, to pretend that I don’t know how the Predictor works. Cognizant of the behind-the-scenes machinations, I will of course choose to take only Box B, because I want that million dollars.

    (To be honest, I’m not sure I entirely understand the concept of random versus non-random choice. If one “chooses randomly,” is one choosing at all? And if we take Sam Harris’s perspective, is it even possible to choose [randomly]?)

    Wikipedia’s comment on the above thinking process:

    “The second strategy suggests taking only B. By this strategy, we can ignore the possibilities that return $0 and $1,001,000, as they both require that the Predictor has made an incorrect prediction, and the problem states that the Predictor is almost never wrong. Thus, the choice becomes whether to receive $1,000 (both boxes) or to receive $1,000,000 (only box B)—so taking only box B is better.

    In his 1969 article, Nozick noted that ‘To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.'”

    That’s precisely what I’m doing: I’m dismissing a priori the possibility of an incorrect prediction. Even if the possibility exists, the probability of an incorrect prediction is minuscule.

    Posted December 22, 2013 at 9:36 am | Permalink
  2. Malcolm says

    Well, in a sense the player can’t lose; he can always just take both and be sure to glom a grand.

    But should you take B alone? After all, by the time you choose, the money’s already in there, or it isn’t. And if you take both, at least you know that you get that thousand.

    Posted December 22, 2013 at 12:17 pm | Permalink