Newcomb's problem
A near-perfect predictor has filled two boxes. Take only the opaque one, or take both? Two airtight arguments give opposite answers.
- Reading time
- 4 min
- Difficulty
- Deep
- Branch
- Decision
- First posed
- William Newcomb, c. 1960; published by Robert Nozick, 1969
The puzzle
In front of you are two boxes. Box A is transparent and contains 1,000 dollars. Box B is opaque. You may take both boxes, or only box B.
Here is the twist. Yesterday, a Predictor, a being with an extraordinary track record, predicted what you would choose. If it predicted you would take only box B, it put 1 million dollars in box B. If it predicted you would take both boxes, it left box B empty. The Predictor has made this prediction for thousands of people, and has been right almost every time. The boxes are now sealed; the prediction has already been made.
What do you do?
Robert Nozick, who published the problem in 1969 after hearing it from the physicist William Newcomb, wrote that almost everyone finds the answer perfectly obvious. The difficulty is that people divide almost evenly, and each side thinks the other is being silly.
Why it matters
Newcomb's problem tests the foundations of decision theory, the theory of rational choice used in economics, game theory and artificial intelligence. Two of the most natural principles of rational choice give opposite recommendations, and it is not obvious which to give up.
It also touches free will and determinism: if your choice can be predicted, is it still a choice? And it has practical cousins in situations where others' expectations of your behaviour affect what they do, such as trust, reputation and commitment.
Key terms
- Decision theoryThe formal study of how to choose rationally under uncertainty, usually by weighing outcomes by their probabilities and values.
- Expected utilityThe value of an option calculated by weighting the utility of each possible outcome by its probability and adding the results. Standard decision theory says to maximise it.
- Dominance principleThe rule that if one option gives a better result than another whatever else is true of the world, you should choose it.
- Causal decision theoryThe theory that you should choose the act whose effects you expect to be best, ignoring correlations your choice does not cause. It recommends taking both boxes in Newcomb's problem.
- Evidential decision theoryThe theory that you should choose the act which, if you learned you had done it, would be the best news about the outcome. It recommends taking one box in Newcomb's problem.
Proposed solutions
Take one box
Take only box B. Those who do almost always walk away with 1 million dollars; those who take both almost always get 1,000.
Take both boxes
Take both. The money is already there or not. Whatever is in box B, you get 1,000 dollars more by taking both.
The problem is incoherent
A perfectly reliable predictor of free choices is impossible, or the scenario conceals a contradiction.
Each solution, tested
Take one box
- How it resolves the paradox
- Look at the evidence. The Predictor is almost always right, so if you one-box, you almost certainly get 1 million dollars, and if you two-box, you almost certainly get 1,000. Evidential decision theory recommends the act that is the best news about the outcome, and one-boxing is overwhelmingly good news. As the slogan goes: if you're so smart, why aren't you rich?
- Why it doesn’t fully work
- Your choice now cannot change what is already in the box. Whatever is there, taking both gets you 1,000 dollars more. One-boxing is like refusing free money because of a superstition that your choice can reach back in time.
- What defenders say
- One-boxers answer that the point is not causation but correlation: you are the kind of person who one-boxes, and the Predictor knew that. Being that kind of person is what pays.
Take both boxes
- How it resolves the paradox
- The dominance principle says that if one option is better than another in every possible state of the world, choose it. Either box B contains 1 million dollars or it is empty. In the first case, taking both gives 1,001,000 dollars rather than 1 million. In the second, 1,000 rather than nothing. Taking both is better either way. Causal decision theory agrees: choose the act whose effects are best, and your choice has no effect on the box's contents.
- Why it doesn’t fully work
- Two-boxers predictably walk away with only 1,000 dollars while one-boxers become millionaires. A theory of rationality that reliably makes you poorer seems to have something wrong with it.
- What defenders say
- Two-boxers answer that the Predictor rewards irrationality: it is like a world where people who believe the moon is made of cheese are given money. Rational people can do badly in situations designed to reward irrationality.
The problem is incoherent
- How it resolves the paradox
- If you are free, nothing could reliably predict your choice before you make it. If something could, then your 'choice' is determined, and asking what you should rationally choose may be confused. Some argue that the problem trades on our inability to imagine the Predictor's mechanism.
- Why it doesn’t fully work
- The problem does not require perfect prediction. A predictor right 90 percent of the time still generates the dilemma. And people are predictable to some degree: psychologists can predict many choices well above chance.
Who argued what
- Take one box Many decision theorists, including Terry Horgan, Most participants in surveys of non-philosophers
- Take both boxes David Lewis, James Joyce, Allan Gibbard
- The problem is incoherent Isaac Levi
Where you stand
What would you take?
In surveys
Choose one to see how people answer.
Variations
The transparent box
Suppose both boxes are transparent. You can see that box B contains 1 million dollars. Do you still take only box B? Most people now say they would take both, but the Predictor, which is almost always right, put the money there because it predicted you would one-box. What does that tell you?
The smoking lesion
Suppose a genetic lesion causes both a desire to smoke and lung cancer, while smoking itself is harmless. Should you refrain from smoking because smoking is evidence you have the lesion? Almost everyone says no. Causal decision theorists use this case to argue that evidential reasoning goes wrong in exactly the way one-boxers do.
Go deeper
How people actually choose
In the 2020 PhilPapers survey of professional philosophers, more respondents favoured two-boxing than one-boxing, though a large share were undecided. In informal polls of the general public, such as one run by the Guardian in 2016 with over 30,000 participants, a majority chose one box. The disagreement between experts and laypeople is itself part of the puzzle's fascination.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Causal Decision Theory
- WikipediaNewcomb's paradox
- SurveyPhilPapers Survey, 2020 · question: Newcomb's problem; about 1,785 professional philosophers; accept or lean toward
- PaperNewcomb's Problem and Two Principles of Choice, Robert Nozick
- BookParadoxes, R. M. Sainsbury
- BookEvidence, Decision and Causality, Arif Ahmed
Open questions
- Can your choice now be evidence about something already fixed?
- Should rationality be judged by its results, or by its reasons?
- If someone could predict you perfectly, would you still be choosing?