Gaming Newcomb’s Paradox I: Problem Solved
One of the many annoying decision theory puzzles is Newcomb’s Paradox. The 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.
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.
The following standard chart shows the possible results:
|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|
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 Predicator 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 Predicator 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.
Gamers of the sort who play Pathfinder, D&D and other such role playing games know how to properly solve this paradox. The Predictor has at least $1,001,000 on hand (probably more, since it will apparently play the game with anyone) and is worth experience points (everything is worth XP). The description just specifies its predictive abilities for the game and no combat abilities are mentioned. So, the solution is to beat down the Predictor, loot it and divide up the money and experience points. It is kind of a jerk when it comes to this game, so there is not really much of a moral concern here.
It might be claimed that the Predictor could not be defeated because of its predictive powers. However, knowing what someone is going to do and being able to do something about it are two very different matters. This is nicely illustrated by the film Billy Jack:
[Billy Jack is surrounded by Posner’s thugs]
Mr. Posner: You really think those Green Beret Karate tricks are gonna help you against all these boys?
Billy Jack: Well, it doesn’t look to me like I really have any choice now, does it?
Mr. Posner: [laughing] That’s right, you don’t.
Billy Jack: You know what I think I’m gonna do then? Just for the hell of it?
Mr. Posner: Tell me.
Billy Jack: I’m gonna take this right foot, and I’m gonna whop you on that side of your face…
[points to Posner’s right cheek]
Billy Jack: …and you wanna know something? There’s not a damn thing you’re gonna be able to do about it.
Mr. Posner: Really?
Billy Jack: Really.
[kicks Posner’s right cheek, sending him to the ground]
So, unless the Predictor also has exceptional combat abilities, the rational solution is the classic “shoot and loot” or “stab and grab.” Problem solved.