Newcomb's paradox

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Newcomb's Paradox is a "paradox" about playing games with an opponent who knows the future. It was created by William Newcomb of the University of California's Lawrence Livermore Laboratory, spread to the philosophical community by Robert Nozick in 1969, and appeared in Martin Gardner's Scientific American column in 1974.

Imagine two players, named Predictor and Chooser, playing the following game. Chooser is presented with two boxes: an open box containing $1000, and a closed box that contains either $1,000,000, or $0 (he doesn't know which). Chooser must decide whether he wants to be given the contents of both boxes, or just the contents of the closed box. Based on this information, a rational Chooser will always take both boxes.

The complication - The day prior, Predictor will predict how Chooser will choose. If he predicts that Chooser will take only the closed box, then he will put $1,000,000 in the closed box. If he predicts that Chooser will take both boxes, he will leave that box empty. The question is: should Chooser take just the closed box or take both boxes?

The analysis of Newcomb's Paradox is often conducted using two different assumptions about Predictor:

  • Predictor can see the future and is 100% accurate
  • Predictor estimates the future based on human behavior

Contents

Predictor can see the future

If Chooser takes the closed box, then it will contain $1,000,000. If Chooser takes both boxes, then the closed box will be empty, and the profit will be only $1,000. Approximately half the population asked this question immediately conclude that the optimal solution is to take the closed box.

The other half the population decide that Chooser should take both boxes reasoning that at the time when Chooser walks up to the boxes, the contents have already been set. The closed box is either empty or full. It's too late for the contents of the boxes to change. Chooser might as well take whatever's in both boxes. Whether the closed box is empty or full, he'll clearly make $1000 more by choosing both boxes than by choosing just one box. Causation only goes forward. Events in the future can't cause results in the past, so there can't be any harm in taking both boxes.

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."

Attempts to resolve the paradox

Philosophers have proposed many solutions to the paradox that avoid backward causation. Some have suggested that a rational person will choose 2, and an irrational person will choose 1, therefore irrational people do better at this game. Others have suggested that if time machines or perfect preditors can exist, then there is no free will and Chooser will do whatever he's fated to do. Others have suggested that the paradox itself shows that it's impossible to ever know the future.

Other people have suggested that in a world with time machines, causation can go backwards. If a person truly knows the future, and that knowledge affects his actions, then events in the future will be causing effects in the past. If causation can go backwards, then the paradox is straightforward. Chooser can freely pick either closed or both. That information will then go back in time and cause the closed box to have been empty or full. It's therefore better to choose the closed box rather than both. If Chooser tries picking both instead, he will later discover that his choice caused that box to have been empty all along, and he'll receive less money. This resolves this form of the paradox.

Glass box

Newcomb's Paradox has been extended with the question of how behaviors would be changed if the closed box is made of glass. Now what should Chooser do? If he sees $1,000,000 in the closed box, then he might as well choose both boxes, and get both the $1,000,000 and the $1,000. If he sees the closed box is empty, he would be angry at being deprived of a chance at the big prize, and so could choose just the 1 box to demonstrate that the game is a fraud. Either way, his actions will be the opposite of what was predicted, which contradicts the assumption that the prediction is always right.

Some philosophers have attempted to argue that this paradox is equivalent to the grandfather paradox. In the grandfather paradox, a person travels back in time, which leads to a chain of events preventing that from happening. In Newcomb's paradox, the information about the choice travels back in time, which leads to a chain of events changing the choice.

The glass box extention of Newcomb's paradox could be taken as a proof that either:

  • it is impossible to know the future
  • that knowledge of the future is only possible in cases where the knowledge itself won't prevent that future
  • that the universe will conspire to prevent self-contradictory causal loops (via the Novikov self-consistency principle, for example). Chooser might accidentally make the wrong selection, or he might misunderstand the rules, or the time machine/prediction engine might break.

Predictor has no special knowledge of the future

Readers of the Scientific American article responded to the paradox in approximately a 5 to 2 ratio in favor of choosing only the closed box. A Predictor working from that data point (and assuming the Chooser is himself a Scientific American reader) would believe that he could achieve about 71% accuracy by always predicting that Chooser will take the closed box. In this case, a rationale Chooser should always take both boxes. The paradox rapidly devolves into an analysis of statistical preferences for risk avoidance and tolerance.

This can be seen more easily if the dollar values are changed. For example, if the amount in the open box is reduced to $1, essentially all Choosers will select the closed box - the incremental value of the dollar does not justify the risk. On the other hand, almost all Choosers will select both boxes if the amount in the open box is raised to $900,000.

References

  • Nozick, Robert (1969), "Newcomb's Problem and Two principles of Choice," in Essays in Honor of Carl G. Hempl, ed. Nicholas Rescher, Synthese Library (Dordrecht, Holland: D. Reidel), p 115.
  • Gardner, Martin (1974), "Mathematical Games," Scientific American, March 1974, p. 102; reprinted with an addendum and annotated bibliography in his book The Colossal Book of Mathematics (ISBN 0-393-02023-1)
  • Campbell, Richmond and Lanning Sowden, ed. (1985), Paradoxes of Rationality and Cooperation: Prisoners' Dilemma and Newcomb's Problem, Vancouver: University of British Columbia Press. (an anthology discussing this paradox, with an extensive bibliography)
  • Levi, Isaac (1982), "A Note on Newcombmania," Journal of Philosophy 79 (1982): 337-42. (a paper discussing the popularity of this paradox)
  • http://members.aol.com/kiekeben/newcomb.html
  • http://slate.msn.com/?id=2061419


References

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