A Philosopher's Blog

Gaming Newcomb’s Paradox II: Mechanics

Posted in Philosophy, Reasoning/Logic by Michael LaBossiere on October 1, 2014
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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.

As a philosopher, a game master (a person who runs a tabletop role playing game) and an author of game adventures, I am rather fond of puzzles and paradoxes. As a philosopher, I can (like other philosophers) engage in the practice known as “just making stuff up.” As an adventure author, I can do the same—but I need to present the actual mechanics of each problem, puzzle and paradox. For example, a trap description has to specific exactly how the trap works, how it may be overcome and what happens if it is set off. I thought it would be interesting to look at Newcomb’s Paradox from a game master perspective and lay out the possible mechanics for it. But first, I will present the paradox and two stock attempts to solve it.

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 laying out some possible mechanics in gamer terms.

One obvious advantage of crafting the mechanics for a game is that the author and the game master know exactly how the mechanic works. That is, she knows the truth of the matter. While the players in role-playing games know the basic rules, they often do not know the full mechanics of a specific challenge, trap or puzzle. Instead, they need to figure out how it works—which often involves falling into spiked pits or being ground up into wizard burger. Fortunately, Newcomb’s Paradox has very simple game mechanics, but many variants.

In game mechanics, the infallible Predictor is easy to model. The game master’s description would be as follows: “have the player character (PC) playing the Predictor’s game make her choice. The Predictor is infallible, so if the player takes box B, she gets the million. If the player takes both, she gets $1,000.” In this case, the right decision is to take B. After all, the Predictor is infallible. So, the solution is easy.

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

A less-than infallible Predictor is also easy to model with dice. The description of the Predictor simply specifies the accuracy of its predictions. So, for example: “The Predictor is correct 99% of the time. After the player character makes her choice, roll D100 (generating a number from 1-100). If you roll 100, the Predictor was wrong. If the PC picked just box B, it is empty and she gets nothing because the Predictor predicted she would take both. If she picked both, B is full and she gets $1,001,000 because the Predictor predicted she would just take one. If you roll 1-99, the Predictor was right. If the PC picked box B, she gets $1,000,000. If she takes both, she gets $1,000 since box B is empty.” In this case, the decision is a gambling matter and the right choice can be calculated by considering the chance the Predictor is right and the relative payoffs. Assuming the Predictor is “almost always right” would make choosing only B the rational choice (unless the player absolutely and desperately needs only $1,000), since the player who picks just B will “almost always” get the $1,000,000 rather than nothing while the player who picks both will “almost always” get just $1,000. But, if the Predictor is “almost always wrong” (or even just usually wrong), then taking both would be the better choice. And so on for all the fine nuances of probability. The solution is relatively easy—it just requires doing some math based on the chance the Predictor is correct in its predictions. As such, if the mechanism of the Predicator is specified, there is no paradox and no problem at all. But, of course, in a role-playing game puzzle, the players should not know the mechanism.

If the game master is doing her job, when the players are confronted by the Predictor, they will not know the predictor’s predictive powers (and clever players will suspect some sort of trick or trap). The game master will say something like “after explaining the rules, the strange being says ‘my predictions are nearly always right/always right’ and sets two boxes down in front of you.” Really clever players will, of course, make use of spells, items, psionics or technology (depending on the game) to try to determine what is in the box and the capabilities of the Predictor. Most players will also consider just attacking the Predictor and seeing what sort of loot it has. So, for the game to be played in accord with the original version, the game master will need to provide plausible ways to counter all these efforts so that the players have no idea about the abilities of the Predictor or what is in box B. In some ways, this sort of choice would be similar to Pascal’s famous Wager: one knows that the Predictor will get it right or it won’t. But, in this case, the player has no idea about the odds of the Predictor being right. In this case, from the perspective of the player who is acting in ignorance, taking both boxes yields a 100% chance of getting $1,000 and somewhere between 0 and 100% chance of getting the extra $1,000,000. Taking the B box alone yields a 100% chance of not getting the $1,000 and some chance between 0% and 100% of getting $1,000,000. When acting in ignorance, the safe bet is to take both: the player walks away with at least $1,000. Taking just B is a gamble that might or might not pay off. The player might walk away with nothing or $1,000,000.

But, which choice is rational can depend on many possible factors. For example, suppose the players need $1,000 to buy a weapon they need to defeat the big boss monster in the dungeon, then picking the safe choice would be the smart choice: they can get the weapon for sure. If they need $1,001,000 to buy the weapon, then picking both would also be a smart choice, since that is the only way to get that sum in this game. If they need $1,000,000 to buy the weapon, then there is no rational way to pick between taking one or both, since they have no idea what gives them the best chance of getting at least $1,000,000. Picking both will get them $1,000 but only gets them the $1,000,000 if the Predictor predicted wrong. And they have no idea if it will get it wrong. Picking just B only gets them $1,000,000 if the Predictor predicted correctly. And they have no idea if it will get it right.

In the actual world, a person playing the game with the Predictor would be in the position of the players in the role-playing game: she does not know how likely it is that the Predictor will get it right. If she believes that the Predictor will probably get it wrong, then she would take both. If she thinks it will get it right, she would take just B. Since she cannot pick randomly (in Nozick’s scenario B is empty if the players decides by chance), that option is not available. As such, Newcomb’s Paradox is an epistemic problem: the player does not know the accuracy of the predictions but if she did, she would know how to pick. But, if it is known (or just assumed) the Predictor is infallible or almost always right, then taking B is the smart choice (in general, unless the person absolutely must have $1,000). To the degree that the Predictor can be wrong, taking both becomes the smarter choice (if the Predictor is always wrong, taking both is the best choice). So, there seems to be no paradox here. Unless I have it wrong, which I certainly do.


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