Newcomb's Paradox
An Analysis
Introduction and Setup
Newcomb’s Paradox is a famous thought experiment in philosophy and decision theory. It goes like this:
You are presented with two boxes.
Box A contains £1,000.
Box B contains either £1,000,000 or nothing. You must choose:
Take only Box B, or
Take both Box A and Box B.
But there is a twist. A predictor - said to be almost infallible - has already made a prediction about what you will choose.
If the predictor foresaw that you would take only Box B, then he placed £1,000,000 in it.
If the predictor foresaw that you would take both boxes, then he left Box B empty.
So the puzzle is: which should you choose?
Counterfactual Nature
Now the premise is, of course, counterfactual. In the real world, there are no people capable of predicting human decisions with one hundred percent accuracy. But thought experiments like this are a kind of game. If we are going to play, we must accept the rules. So let us, for the sake of argument, assume that such prediction is possible and consider what follows.
There are a number of variations in the way the problem is commonly presented. Let us assume, to begin with, that the predictor is 100% accurate.
Opposing Arguments
At first glance, two seemingly reasonable arguments pull in opposite directions. On the one hand, if the predictor is always right, then one-boxing guarantees £1 million, while two-boxing guarantees only £1,000. On the other hand, many object that your choice now cannot possibly affect what the predictor has already placed in the boxes. So whatever is in Box B, why not take both and gain the extra £1,000? This is the crux of the paradox.
One-Boxing as the Rational Choice
I argue for one-boxing. The argument in favour is straightforward. One-boxing guarantees you a million pounds; two-boxing leaves you with only a thousand.
The objections to this position all rest on denying the possibility of backward causation. “Your choice can’t affect what’s already in the box” say the two-boxers.
That is true - in the real world. But this isn’t the real world. This is the world of Newcomb’s Paradox, and in the game-world of the paradox the future can affect the past. This is not an additional assumption but is implied by the very setup of the problem. If perfect prediction of the future is possible, then the future must already constrain the past. To see this, consider an example.
Suppose a person asks a predictor, “Should I take a voyage next week?” The predictor replies, “No. The ship is going to sink, and you will probably die.” A week later, the ship sets sail, hits an iceberg, and sinks. The predictor’s vision of the sinking one week earlier exists precisely because the sinking occurs. The disaster in the future causes the vision in the past. That is what perfect prediction entails. If the ship hadn’t sunk, the predictor would not have said it was going to. So if we accept 100% accuracy, then backward causation is part of the rules. According to those rules, one-boxing is the clear choice. This is all very silly of course, but denying this within the scenario is simply refusing to play by the rules of the game.
Less Than Perfect Prediction
If prediction is less than 100% accurate, then the situation changes. Assuming the predictor is correct more than 50 % of the time (and they’re not much of a predictor if they are not!), then the future is still having some effect on the past. In this setup standard game theory can be applied. In such cases, one-boxing will usually remain the best strategy, because the expected payoff is larger. The exact choice depends on the particular probabilities involved, which we are not given in the standard presentation of the paradox.
If we do not know the predictor’s reliability at all - if it is shrouded in uncertainty - then Bayesian game theory provides the appropriate tools. We would assign prior probabilities to different levels of predictor accuracy and update them as evidence accumulates.
Final Reflection
In the end, however, there is another possible answer: to dismiss the problem altogether. Not only is the situation in Newcomb’s Paradox counterfactual, it is in fact impossible, for it requires paradoxical conditions such as flawless prediction and backward causation. It is not difficult to construct a scenario leading to a logical paradox where the predictor causes a change in the future, which in turn causes a change in the prediction, which in turn causes another change in the future, and so on and on. If one insists on strict realism, then the whole setup collapses. But if one treats it as an intellectual game, then the analysis given here shows that one-boxing is the option most consistent with the rules of the puzzle as stated.