In philosophy and mathematics, Newcomb's problem, also known as Newcomb's paradox, is a thought experiment posing a decision problem in which a player must decide whether to choose one of two or both boxes under conditions in which a being, often called the "predictor", decides in advance what they contain, and is able to predict the player's choices with near-certainty.
Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. It was first analyzed in a philosophy paper by Robert Nozick in 1969 and appeared in the March 1974 issue of Scientific American, in Martin Gardner's "Mathematical Games". Reprinted with an addendum and annotated bibliography in his book The Colossal Book of Mathematics (). Today it is a much debated problem in the philosophical branch of decision theory.
Definition A + B(B has $0) B(B has $1,000,000) A + B $1,000 $1,001,000 B $0 $1,000,000 In the standard version of Newcomb's problem, two boxes are 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 transparent, or open, and always contains a visible $1,000. Box B is opaque, or closed, and its content has already been set by the predictor: If the predictor has predicted that 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 player does not know what the predictor predicted or what box B contains while making the choice.
Limit case A + B B A + B $1,000 Impossible B Impossible $1,000,000
In the limit case of Newcomb's problem, the predictor is not only extremely reliable but actually infallible. Many solutions focus on the limit case, especially as there can seem to be no decision-relevant difference between extremely high probability and certainty. However, Nozick defended one-boxing in the limit case and two-boxing in other cases.
Relationship to game theory Some treatments of Newcomb's problem attempt to interpret it in terms of game theory, with the predictor interpreted as a second player. However, the problem does not qualify as a game in the sense defined by game theory, since the predictor's choice is probabilistically dependent on the player's choice, and the predictor has no preference order over the possible outcomes.
Strategies 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." The problem continues to divide philosophers today.Bourget, D., Chalmers, D. J. (2014). "What do philosophers believe?" Philosophical Studies, 170(3), 465–500. In a 2020 survey, a modest plurality of professional philosophers chose to take both boxes (39.0% versus 31.2%).
Decision theory offers two strategies for this game that rely on different principles: the expected utility principle and the strategic dominance principle. The problem is sometimes considered a paradox (although this classification is disputed) because two seemingly logical analyses yield conflicting answers regarding which choice maximizes the player's payout. Expected utility: Due to the high accuracy of the predictor, it can seem that it is not realistically open to the player (or in the limit case, open at all to the player) to make a choice that the predictor did not predict. Considering the expected utility in such conditions, the player should choose to take only box B. This choice, called one-boxing, statistically maximizes the player's winnings, resulting in approximately $1,000,000 per game. (This argument from expected utility was given by Nozick and echoed by other one-boxers such as Horgan, but Huemer believes that a correct interpretation of expected-utility maximization would support two-boxing.) Strategic dominance: Since the predictor has already made its prediction, it can seem that choosing both boxes A and B will always yield $1,000 more than only choosing B, and that if this turns out to equal exactly $1,000, then the option of getting a million from box B was simply not open to the player. Under the strategic dominance principle, the player should then choose the strategy that is always better, and take both boxes (two-boxing).
David Wolpert and Gregory Benford point out that paradoxes arise when not all relevant details of a problem are specified, and there is more than one "intuitively obvious" way to fill in those missing details. They suggest that, in Newcomb's paradox, the debate over which strategy is "obviously correct" stems from the fact that interpreting the problem details differently can lead to two distinct noncooperative games. Each strategy is optimal for one interpretation of the game but not the other. They then derive the optimal strategies for both of the games, which turn out to be independent of the predictor's infallibility, questions of causality, determinism, and free will. A similar analysis of different interpretations of the problem is given in below.
Newcomb's problem has been used as an argument against causal decision theory, on the premise that causal decision theory supports taking both boxes in Newcomb's problem, but taking only one box is the superior choice (on expected-utility grounds). Caspar Oesterheld and Vincent Conitzer, for instance, have argued against the use of causal decision theory when applied to Newcomb's paradox. They argue that there exist multiple scenarios where the use of causal decision theory causes the agent to voluntarily lose money, and that causal decision theorists are subject to diachronic Dutch books.
Interpretations Different interpretations as to why the predictor is so accurate yield different rational solutions to the problem. Some such interpretations are illustrated in this section with diagrams in a convention similar to Mackie's 1977 paper "Newcomb's Paradox and the Direction of Causation", using white squares for events whose outcome is "open" to the player's rational deliberation, black squares for events whose outcome is not open to deliberation, and arrows for causal connections. In the diagrams, references to a choice being "free" refer to the choice being open to influence by decision-theoretical reasoning about the rational choice in the decision problem, not necessarily in some broader "free will" sense, although some cited authors have discussed such broader conceptions.
Causal connection
If (a) the predictor is always putting $1,000,000 in the opaque box and then using a hidden trapdoor to remove it if the player goes for the transparent box, it is reasonable to one-box. Similarly, if retrocausality (backwards causation) is possible (b), or if (c) the problem situation is played multiple times so that the predictor learns from the player's behavior in one round of play to make his prediction for the next round, it is clearly reasonable to one-box. In cases (a)–(c), there is a causal connection between two-boxing and losing the million, so there is no controversy. Variants of the retrocausality include that the current post-determination premise may be merely a lie or illusion; it may in fact be the test that is used to determine the contents.
William Lane Craig has suggested that, in a world with perfect predictors (or time machines, because a time machine could be used as a mechanism for making a prediction), retrocausality can occur. The chooser's choice can be said to have caused the predictor's prediction. Some have concluded that if time machines or perfect predictors can exist, then there can be no free will and choosers will do whatever they are fated to do. Taken together, the paradox is a restatement of the old contention that free will and determinism are incompatible, since determinism enables the existence of perfect predictors. Put another way, this paradox can be equivalent to the grandfather paradox; the paradox presupposes a perfect predictor, implying the "chooser" is not free to choose, yet simultaneously presumes a choice can be debated and decided. This suggests to some that the paradox is an artifact of these contradictory assumptions.
If the predictor is posited as infallible and incapable of error, it can seem that it is impossible for there to be a true answer as to what it is most reasonable to choose, as there is no true choice, and the player's decision is predetermined by the psychological facts surveyed by the predictor. For instance, if the predictor has such an accurate track record because (f) he is a capable hypnotist who manipulates all choosers to one-box or two-box in accordance with whether he chose to put the money in the opaque box, then it would be reasonable to two-box if one could, but by construction, one is not free to. Similarly, if the predictor has an infallibly correct psychology (g), then the player's choice cannot be determined by a strategist's recommendation of what is most reasonable to do, as it is fixed in advance by his psychological character. Any solution then becomes moot. Nozick avoids this issue by positing that the predictor's predictions are "almost certainly" correct, thus sidestepping any issues of infallibility and causality. Nozick also stipulates that if the predictor predicts that the player will choose randomly, then box B will contain nothing. This assumes that inherently random or unpredictable events would not come into play anyway during the process of making the choice, such as free will or quantum mind processes.
This interpretation of Newcomb's paradox is related to logical fatalism in that they both suppose absolute certainty of the future. In logical fatalism, this assumption of certainty creates circular reasoning ("a future event is certain to happen, therefore it is certain to happen"), while Newcomb's paradox considers whether the participants of its game are able to affect a predestined outcome..
Character formation