Newcomb's paradox
Encyclopedia
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.
, as a superintelligent alien, as a deity
, as a brain-scanning computer, 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.
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."
If the player believes that the predictor can correctly predict any thoughts he or she will have, but has access to some source of random numbers that the predictor cannot predict (say, a coin to flip, or a quantum process), then the game depends on how the predictor will react to (correctly) knowing that the player will use such a process. If the predictor predicts by reproducing the player's process, then the player should open both boxes with 1/2 probability and will receive an average of $251,000; if the predictor predicts the most probable player action, then the player should open both with 1/2 - epsilon probability and will receive an average of ~$500,999.99; and if the predictor places $0 whenever they believe that the player will use a random process, then the traditional "paradox" holds unchanged.
The use of first person in the formulation of the second argument is essential:
only when playing the role of the chooser I feel that I determine the fate of the past event. Looking from aside at another person participating in the experiment does not arouse a feeling of contradiction.
His choice and its prediction are part of a causal chain, that in principle is not problematic.
A solution of the paradox must point out an error in one of the two arguments. Either the intuition is wrong, or there is something wrong with the way proposed for affecting the past.
) that is equivalent to the paradox. It is this:
Suppose that the omnipotent predictor predicted the grade I will get in tomorrow's exam, and wrote his prediction in a note. Since the content of the note was determined a while ago, I cannot change it. Since I believe that it reflects precisely the grade I will get, I cannot also change my grade. So I can just as well rest, rather than prepare for the exam (hence the name "the idle argument").
In both situations an equivalence between a past event P and a future event F is used to draw a paradoxical conclusion, and both use the same argumentation. In Newcomb's paradox the claim is "I can determine F, hence I can change P", while in the idle argument the claim is "I cannot change P, hence I cannot determine F", which is the same argument, formulated in reverse direction.
will lead one to believe that one should expect the most utility
(or money) from taking only box B. However if one uses the Dominance
principle, one would expect to benefit most from taking both boxes.
More recent work has reformulated the problem as a noncooperative game in which players set the conditional distributions in a Bayes net. It is straight-forward to prove that the two strategies for which boxes to choose make mutually inconsistent assumptions for the underlying Bayes net. Depending on which Bayes net one assumes, one can derive either strategy as optimal. In this there is no paradox, only unclear language that hides the fact that one is making two inconsistent assumptions.
Some argue that Newcomb's Problem is a paradox
because it leads logically to self-contradiction. Reverse causation
is defined into the problem and therefore logically there can be no free will
. However, free will is also defined in the problem; otherwise the chooser is not really making a choice.
Other philosophers have proposed many solutions to the problem, many eliminating its seemingly paradoxical nature:
Some suggest a rational person will choose both boxes, and an irrational person will choose just the one, therefore rational people fare better, since the Predictor cannot actually exist. Others have suggested that an irrational person will do better than a rational person and interpret this paradox as showing how people can be punished for making rational decisions.
Others have suggested that in a world with perfect predictors (or time machine
s because a time machine could be the mechanism for making the prediction) 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. Chooser's choice will have already caused Predictor's action. Some have concluded that if time machines or perfect predictors can exist, then there can be no free will
and Chooser will do whatever he's fated to do. Others conclude that the paradox shows that it is impossible to ever know the future. 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. Some philosophers argue this paradox is equivalent to the grandfather paradox
. Put another way, 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. Nozick's exposition specifically excludes backward causation (such as time travel) and requires only that the predictions be of high accuracy, not that they are absolutely certain to be correct. So the considerations just discussed are irrelevant to the paradox as seen by Nozick, which focuses on two principles of choice, one probabilistic and the other causal - assuming backward causation removes any conflict between these two principles.
Newcomb's paradox can also be related to the question of machine consciousness, specifically if a perfect simulation of a person's brain will generate the consciousness of that person. Suppose we take the Predictor to be a machine that arrives at its prediction by simulating the brain of the Chooser when confronted with the problem of which box to choose. If that simulation generates the consciousness of the Chooser, then the Chooser cannot tell if he is standing in front of the boxes in the real world or in the virtual world generated by the simulation. The "virtual" Chooser would thus tell the Predictor which choice the "real" Chooser is going to make.
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
involving a game between two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
is disputed.
Newcomb's paradox was created by William Newcomb
William Newcomb
William Newcomb , a professor and theoretical physicist at the University of California's Lawrence Livermore Laboratory, is best known as the creator of Newcomb's paradox, devised in 1960...
of the University of California
University of California
The University of California is a public university system in the U.S. state of California. Under the California Master Plan for Higher Education, the University of California is a part of the state's three-tier public higher education system, which also includes the California State University...
's Lawrence Livermore Laboratory. However, it was first analyzed and was published in a philosophy paper spread to the philosophical community by Robert Nozick
Robert Nozick
Robert Nozick was an American political philosopher, most prominent in the 1970s and 1980s. He was a professor at Harvard University. He is best known for his book Anarchy, State, and Utopia , a right-libertarian answer to John Rawls's A Theory of Justice...
in 1969, and appeared in Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...
's Scientific American
Scientific American
Scientific American is a popular science magazine. It is notable for its long history of presenting science monthly to an educated but not necessarily scientific public, through its careful attention to the clarity of its text as well as the quality of its specially commissioned color graphics...
column in 1974. Today it is a much debated problem in the philosophical branch of decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
but has received little attention from the mathematical side.
The problem
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 psychicPsychic
A psychic is a person who professes an ability to perceive information hidden from the normal senses through extrasensory perception , or is said by others to have such abilities. It is also used to describe theatrical performers who use techniques such as prestidigitation, cold reading, and hot...
, as a superintelligent alien, as a deity
Deity
A deity is a recognized preternatural or supernatural immortal being, who may be thought of as holy, divine, or sacred, held in high regard, and respected by believers....
, as a brain-scanning computer, etc. However, the original discussion by Nozick says only that the Predictor's predictions are "almost certainly
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
" 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.
| Predicted choice | Actual choice | Payout |
|---|---|---|
| A and B | A and B | $1,000 |
| A and B | B only | $0 |
| B only | A and B | $1,001,000 |
| B only | B only | $1,000,000 |
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.
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."
If the player believes that the predictor can correctly predict any thoughts he or she will have, but has access to some source of random numbers that the predictor cannot predict (say, a coin to flip, or a quantum process), then the game depends on how the predictor will react to (correctly) knowing that the player will use such a process. If the predictor predicts by reproducing the player's process, then the player should open both boxes with 1/2 probability and will receive an average of $251,000; if the predictor predicts the most probable player action, then the player should open both with 1/2 - epsilon probability and will receive an average of ~$500,999.99; and if the predictor places $0 whenever they believe that the player will use a random process, then the traditional "paradox" holds unchanged.
The crux of the problem
The crux of the paradox is in the existence of two contradictory arguments, both being seemingly correct.- A powerful intuitive belief, that past events cannot be affected. My future action cannot determine the fate of an event that happened before the action.
- Newcomb proposes a way of doing precisely this - affecting a past event. The prediction of the Predictor establishes equivalence between my choice (of renouncing the open box) and the content of the closed box, which was determined in the past. Since I can affect the future event, I can also affect the past event, which is equivalent to it.
The use of first person in the formulation of the second argument is essential:
only when playing the role of the chooser I feel that I determine the fate of the past event. Looking from aside at another person participating in the experiment does not arouse a feeling of contradiction.
His choice and its prediction are part of a causal chain, that in principle is not problematic.
A solution of the paradox must point out an error in one of the two arguments. Either the intuition is wrong, or there is something wrong with the way proposed for affecting the past.
The relationship to the idle argument
There is a version of the famous idle argument (see fatalismFatalism
Fatalism is a philosophical doctrine emphasizing the subjugation of all events or actions to fate.Fatalism generally refers to several of the following ideas:...
) that is equivalent to the paradox. It is this:
Suppose that the omnipotent predictor predicted the grade I will get in tomorrow's exam, and wrote his prediction in a note. Since the content of the note was determined a while ago, I cannot change it. Since I believe that it reflects precisely the grade I will get, I cannot also change my grade. So I can just as well rest, rather than prepare for the exam (hence the name "the idle argument").
In both situations an equivalence between a past event P and a future event F is used to draw a paradoxical conclusion, and both use the same argumentation. In Newcomb's paradox the claim is "I can determine F, hence I can change P", while in the idle argument the claim is "I cannot change P, hence I cannot determine F", which is the same argument, formulated in reverse direction.
Attempted resolutions
Many argue that the paradox is primarily a matter of conflicting decision making models. Using the expected utility hypothesisExpected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...
will lead one to believe that one should expect the most utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
(or money) from taking only box B. However if one uses the Dominance
Dominance (game theory)
In game theory, strategic dominance occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play...
principle, one would expect to benefit most from taking both boxes.
More recent work has reformulated the problem as a noncooperative game in which players set the conditional distributions in a Bayes net. It is straight-forward to prove that the two strategies for which boxes to choose make mutually inconsistent assumptions for the underlying Bayes net. Depending on which Bayes net one assumes, one can derive either strategy as optimal. In this there is no paradox, only unclear language that hides the fact that one is making two inconsistent assumptions.
Some argue that Newcomb's Problem is a paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
because it leads logically to self-contradiction. Reverse causation
Retrocausality
Retrocausality is any of several hypothetical phenomena or processes that reverse causality, allowing an effect to occur before its cause....
is defined into the problem and therefore logically there can be no free will
Free will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...
. However, free will is also defined in the problem; otherwise the chooser is not really making a choice.
Other philosophers have proposed many solutions to the problem, many eliminating its seemingly paradoxical nature:
Some suggest a rational person will choose both boxes, and an irrational person will choose just the one, therefore rational people fare better, since the Predictor cannot actually exist. Others have suggested that an irrational person will do better than a rational person and interpret this paradox as showing how people can be punished for making rational decisions.
Others have suggested that in a world with perfect predictors (or time machine
Time travel
Time travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...
s because a time machine could be the mechanism for making the prediction) 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. Chooser's choice will have already caused Predictor's action. Some have concluded that if time machines or perfect predictors can exist, then there can be no free will
Free will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...
and Chooser will do whatever he's fated to do. Others conclude that the paradox shows that it is impossible to ever know the future. Taken together, the paradox is a restatement of the old contention that free will and determinism
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...
are incompatible, since determinism enables the existence of perfect predictors. Some philosophers argue this paradox is equivalent to the grandfather paradox
Grandfather paradox
The grandfather paradox is a proposed paradox of time travel first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent . The paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler's...
. Put another way, 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. Nozick's exposition specifically excludes backward causation (such as time travel) and requires only that the predictions be of high accuracy, not that they are absolutely certain to be correct. So the considerations just discussed are irrelevant to the paradox as seen by Nozick, which focuses on two principles of choice, one probabilistic and the other causal - assuming backward causation removes any conflict between these two principles.
Newcomb's paradox can also be related to the question of machine consciousness, specifically if a perfect simulation of a person's brain will generate the consciousness of that person. Suppose we take the Predictor to be a machine that arrives at its prediction by simulating the brain of the Chooser when confronted with the problem of which box to choose. If that simulation generates the consciousness of the Chooser, then the Chooser cannot tell if he is standing in front of the boxes in the real world or in the virtual world generated by the simulation. The "virtual" Chooser would thus tell the Predictor which choice the "real" Chooser is going to make.
External links
- Newcomb's Paradox by Franz Kiekeben
- Thinking Inside the Boxes by Jim Holt, for SlateSlate (magazine)Slate is a US-based English language online current affairs and culture magazine created in 1996 by former New Republic editor Michael Kinsley, initially under the ownership of Microsoft as part of MSN. On 21 December 2004 it was purchased by the Washington Post Company...
- Free Will: Two Paradoxes of Choice (lecture) by Roderick T. Long
- Newcomb's Problem by Marion Ledwig
- Newcomb's Problem and Regret of Rationality by Eliezer Yudkowsky
- The Resolution of Newcomb's Paradox by Chris Langan