Gaming Newcomb’s Paradox III: What You Actually Decide
Newcomb’s Paradox was created by William Newcomb of the University of California’s Lawrence Livermore Laboratory. The dread philosopher Robert Nozick published a paper on it in 1969 and it was popularized in Martin Gardner’s 1972 Scientific American column.
In this essay I will present the game that creates the paradox and then discuss a specific aspect of Nozick’s version, namely his stipulation regarding the effect of how the player of the game actually decides.
The paradox involves a game controlled by the Predictor, a being that is supposed to be masterful at predictions. Like many entities with but one ominous name, the Predictor’s predictive capabilities vary with each telling of the tale. The specific range is from having an exceptional chance of success to being infallible. The basis of the Predictor’s power also vary. In the science-fiction variants, it can be a psychic, a super alien, or a brain scanning machine. In the fantasy versions, the Predictor is a supernatural entity, such as a deity. In Nozick’s telling of the tale, the predictions are “almost certainly” correct and he stipulates that “what you actually decide to do is not part of the explanation of why he made the prediction he made”.
Once the player confronts the Predictor, the game is played as follows. The Predictor points to two boxes. Box A is clear and contains $1,000. Box B is opaque. The player has two options: just take box B or take both boxes. The Predictor then explains to the player the rules of its game: the Predictor has already predicted what the player will do. If the Predictor has predicted that the player will take just B, B will contain $1,000,000. Of course, this should probably be adjusted for inflation from the original paper. If the Predictor has predicted that the player will take both boxes, box B will be empty, so the player only gets $1,000. In Nozick’s version, if the player chooses randomly, then box B will be empty. The Predictor does not inform the player of its prediction, but box B is either empty or stuffed with cash before the players actually picks. The game begins and ends when the player makers her choice.
This paradox is regarded as a paradox because the two stock solutions are in conflict. The first stock solution is that the best choice is to take both boxes. If the Predictor has predicted the player will take both boxes, the player gets $1,000. If the Predicator has predicted (wrongly) that the player will take B, she gets $1,001,000. If the player takes just B, then she risks getting $0 (assuming the Predicator predicted wrong).
The second stock solution is that the best choice is to take B. Given the assumption that the Predictor is either infallible or almost certainly right, then if the player decides to take both boxes, she will get $1,000. If the player elects to take just B, then she will get $1,000,000. Since $1,000,000 is more than $1,000, the rational choice is to take B. Now that the paradox has been presented, I can turn to Nozick’s condition that “what you actually decide to do is not part of the explanation of why he made the prediction he made”.
This stipulation provides some insight into how the Predictor’s prediction ability is supposed to work. This is important because the workings of the Predictor’s ability to predict are, as I argued in my previous essay, rather significant in sorting out how one should decide.
The stipulation mainly serves to indicate how the Predicator’s ability does not work. First, it would seem to indicate that the Predictor does not rely on time travel—that is, it does not go forward in time to observe the decision and then travel back to place (or not place) the money in the box. After all, the prediction in this case would be explained in terms of what the player decided to do. This still leaves it open for the Predictor to visit (or observe) a possible future (or, more accurately, a possible world that is running ahead of the actual world in its time) since the possible future does not reveal what the player actually decides, just what she decides in that possible future. Second, this would seem to indicate that the Predictor is not able to “see” the actual future (perhaps by being able to perceive all of time “at once” rather than linearly as humans do). After all, in this case it would be predicting based on what the player actually decided. Third, this would also rule out any form of backwards causation in which the actual choice was the cause of the prediction. While there are, perhaps, other specific possibilities that are also eliminated, the gist is that the Predictor has to, by Nozick’s stipulation, be limited to information available at the time of the prediction and not information from the future. There are a multitude of possibilities here.
One possibility is that the Predictor is telepathic and can predict based on what it reads regarding the player’s intentions at the time of the prediction. In this case, the best approach would be for the player to think that she will take one box, and then after the prediction is made, take both. Or, alternatively, use some sort of drugs or technology to “trick” the Predictor. The success of this strategy would depend on how well the player can fool the Predictor. If the Predictor cannot be fooled or is unlikely to be fooled then the smart strategy would be to intend to take box B and then just take box B. After all, if the Predictor cannot be fooled, then box B will be empty if the player intends on taking both.
Another possibility is that the Predictor is a researcher—it gathers as much information as it can about the player and makes a shrewd guess based on that information (which might include what the player has written about the paradox). Since Nozick stipulates that the Predictor is “almost certainly” right, the Predictor would need to be an amazing researcher. In this case, the player’s only way to mislead the Predictor is to determine its research methods and try to “game” it so the Predictor will predict that she will just take B, then actually decide to take both. But, once again, the Predictor is stipulated to be “almost certainly” right—so it would seem that the player should just take B. If B is empty, then the Predictor got it wrong, which would “almost certainly” not happen. Of course, it could be contended that since the player does not know how the Predictor will predict based on its research (the player might not know what she will do), then the player should take both. This, of course, assumes that the Predictor has a reasonable chance of being wrong—contrary to the stipulation.
A third possibility is that the Predictor predicts in virtue of its understanding of what it takes to be a determinist system. Alternatively, the system might be a random system, but one that has probabilities. In either case, the Predictor uses the data available to it at the time and then “does the math” to predict what the player will decide.
If the world really is deterministic, then the Predictor could be wrong if it is determined to make an error in its “math.” So, the player would need to predict how likely this is and then act accordingly. But, of course, the player will simply act as she is determined to act. If the world is probabilistic, then the player would need to estimate the probability that the Predictor will get it right. But, it is stipulated that the Predictor is “almost certainly” right so any strategy used by the player to get one over on the Predictor will “almost certainly” fail, so the player should take box B. Of course, the player will do what “the dice say” and the choice is not a “true” choice.
If the world is one with some sort of metaphysical free will that is in principle unpredictable, then the player’s actual choice would, in principle, be unpredictable. But, of course, this directly violates the stipulation that the Predictor is “almost certainly” right. If the player’s choice is truly unpredictable, then the Predictor might make a shrewd/educated guess, but it would not be “almost certainly” right. In that case, the player could make a rational case for taking both—based on the estimate of how likely it is that the Predictor got it wrong. But this would be a different game, one in which the Predictor is not “almost certainly” right.
This discussion seems to nicely show that the stipulation that “what you actually decide to do is not part of the explanation of why he made the prediction he made” is a red herring. Given the stipulation that the Predictor is “almost certainly” right, it does not really matter how its predictions are explained. The stipulation that what the player actually decides is not part of the explanation simply serves to mislead by creating the false impression that there is a way to “beat” the Predictor by actually deciding to take both boxes and gambling that it has predicted the player will just take B. As such, the paradox seems to be dissolved—it is the result of some people being misled by one stipulation and not realizing that the stipulation that the Predictor is “almost certainly” right makes the other irrelevant.