A person is playing a game operated by the Predictor, an entity somehow presented as being exceptionally skilled at predicting people's actions. The exact nature of the Predictor varies between retellings of the paradox. Some assume that the character always has a reputation for being completely infallible and incapable of error; others assume that the predictor has a very low error rate. The Predictor can be presented as a psychic, as a superintelligent alien, as a deity, as a brain-scanning computer, etc. ...So which do you choose?
The player of the game is presented with two boxes, one transparent (labeled A) and the other opaque (labeled B). The player is permitted to take the contents of both boxes, or just the opaque box B. Box A contains a visible $1,000. The contents of box B, however, are determined as follows: At some point before the start of the game, the Predictor makes a prediction as to whether the player of the game will take just box B, or both boxes. If the Predictor predicts that both boxes will be taken, then box B will contain nothing. If the Predictor predicts that only box B will be taken, then box B will contain $1,000,000.
By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents of box B, the fact that its contents are based on the Predictor's prediction, and knowledge of the Predictor's infallibility. The only information withheld from the player is what prediction the Predictor made, and thus what the contents of box B are.
The paradox is that there are two lines of reasoning which both appear to be "rational," yet produce opposite conclusions. The first is this: since the Predictor has already placed the money in the boxes, my decision has no effect on whether or not B has $1,000,000. I should choose both boxes, since that will give me more money than if I only chose one. The second line of reasoning is this: the Predictor is never (or almost never) wrong, so I should choose just box B, because if I choose box B then it will be $1,000,000.
A couple of interesting observations. First, it seems to me to make all the difference in the world whether or not you're the one actually making the choice. If I were observing someone else make the choice, I would think, "Gosh, I hope for his sake that he's the type to take just box B, because then surely he will make more money." But if I'm the one who has to choose, well, it's hard to make such a statement about myself.
There's an interesting discussion about the paradox here, although it's a bit long. The perspective taken there is an appealing one, in my view: rationality ought to win. It's no excuse for the rationalist to say, "I choose both because this is the most logical answer; I cannot help the fact that I am rational and therefore wasn't given the option of making $1,000,000." If that's what it means to be rational to you, then rationality isn't going to get you very far.
Another interesting question: what if the Predictor tells you the money is for charity? Does that change the rationalist's reasoning? Logically speaking, it shouldn't.
I'm adding this to my list of lovable paradoxes, because yet again we have an example demonstrating how reason is insufficient for making decisions. Sometimes logic is an obstacle to being rational.