Newcomb's paradox: Encyclopedia II - Newcomb's paradox - Thoughts on the paradox

Newcomb's paradox - Thoughts on the paradox

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 the closed 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.

The rationality of the person who chooses the closed box depends upon facts concerning the Predictor. If, as posited, the Predictor is 100% accurate, and is completely reliable to put the million dollars in the closed box, and the chooser knows this, then the only rational choice is to pick the closed box. If the players knows the Predictor is unreliable, then the only rational choice is both boxes.

Others have suggest that in a world with perfect predictors (or time machines 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 perfect predictors require determinism. Some philosophers argue this paradox is equivalent to the grandfather paradox.

Newcomb's paradox - Glass box

Newcomb's Problem has been extended with the question of how behaviors would be changed if the closed box is made of glass. Now what should chooser do?

If he sees $1,000,000 in the closed box, then he might as well choose both boxes, and get both the $1,000,000 and the $1,000. If he sees the closed box is empty, he might be angry at being deprived of a chance at the big prize and so choose just the one box to demonstrate that the game is a fraud. Either way, his actions will be the opposite of what was predicted, which contradicts the premise that the prediction is always right.

Some philosophers take the glass box version of Newcomb's paradox as a proof that:

1. It is impossible to know the future.
2. Knowledge of the future is only possible in cases where the knowledge itself won't prevent that future.
3. The universe will conspire to prevent self-contradictory causal loops (via the Novikov self-consistency principle, for example).
4. Chooser might accidentally make the wrong selection, or he might misunderstand the rules, or the time machine/prediction engine might break.

Newcomb's paradox - Predictor has no special knowledge of the future

Suppose Predictor does not have special knowledge of the future and Chooser knows this. A game theory analysis for the case of multiple rounds with memory is straightforward.

If Chooser wants to maximize profit and Predictor wants to maximize the accuracy of his predictions, Chooser should consistently choose only the closed box. However, if Chooser defects from that strategy and chooses both boxes, he will benefit in that round but Predictor will be wrong and will probably retaliate. Nash equilibria (where any defection from the chosen strategies is no benefit) exist when the Chooser always takes 2 boxes and Predictor always predicts that 2 boxes will be chosen (this gives a payout of $1,000 and a perfect prediction every time), or when both choose only the closed box (payout $1,000,000 prediction 100%). An intelligent chooser would attempt to move from the first equilibrium to the second.

Now consider a different case: Predictor does not have special knowledge of the future, but Chooser believes he does. Readers of the Scientific American article responded to the paradox in approximately a 5 to 2 ratio in favor of choosing only the closed box. A Predictor working from that data point (and assuming the Chooser is himself a Scientific American reader) would believe that he could achieve about 71% accuracy by always predicting that Chooser will take the closed box.

In this case, the problem rapidly devolves into an analysis of statistical preferences for risk avoidance and tolerance. This can be seen more easily if the dollar values are changed. For example, if the amount in the open box is reduced to $1, essentially all Choosers will select the closed box - the incremental value of the dollar does not justify the risk. On the other hand, almost all Choosers will select both boxes if the amount in the open box is raised to $900,000.

Adapted from the Wikipedia article "Thoughts on the paradox", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki