Newcomb's paradox, also referred to as Newcomb's problem, is a thought experiment involving a game between two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox is disputed.
Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. However, it was first analyzed and was published in a philosophy paper spread to the philosophical community by Robert Nozick in 1969, and appeared in Martin Gardner's Scientific American column in 1974. Today it is a much debated problem in the philosophical branch of decision theory but has received little attention from the mathematical side.
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. The Predictor can be presented as a psychic, as a superintelligent alien, as a deity, etc. However, the original discussion by Nozick says only 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.
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.
The problem is called a paradox because two strategies that both sound intuitively logical give conflicting answers to the question of what choice maximizes the player's payout. The first strategy argues that, regardless of what prediction the Predictor has made, taking both boxes yields more money. That is, if the prediction is for both A and B to be taken, then the player's decision becomes a matter of choosing between $1,000 (by taking A and B) and $0 (by taking just B), in which case taking both boxes is obviously preferable. But, even if the prediction is for the player to take only B, then taking both boxes yields $1,001,000, and taking only B yields only $1,000,000—taking both boxes is still better, regardless of which prediction has been made.
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