After writing about Simpson's paradox a few weeks ago, and receiving such positive responses to it, I have been asked to write another article on a mathematical paradox. This time the topic of discussion is Newcomb's paradox.

Suppose that there is a game show in which you, the player, are shown two boxes labelled A and B. Box A will always contain \$1000. Box B may contain a million dollars or it may contain absolutely nothing, which is decided before the game begins. The player is given a choice: they can pick only box A or the can pick both box A and box B, and either way they get to keep all of the money in the box or boxes they have chosen. It would seem that the answer is obvious – they player should always select both boxes and will average just over a half a million dollars in prizes.

Except this game has a supernatural twist to it. The contents of box B is decided before the game begins by a superbeing who can predict with perfect accuracy which option the player will select. If the play is greedy and takes both boxes, the being leaves box B empty. If the player only takes the first box, then the being puts the million dollar prize in the second one. And so no matter what the player does, they always receive the exact same prize.

However the player tries to outsmart the game. They go into the game planning to pick box A alone, so that the million dollar prize is added to box B. Then at the last moment the player switches tactics and selects both boxes, and gets the million dollar prize. The superbeing cannot change the prediction, and so the play wins it all.

This is Newcomb's paradox. It is a fun piece of recreational philosophy, but in truth it has two obvious solutions.

Solution 1: The statement of the paradox is itself inconsistent. The superbeing can only make perfect predictions if the player lacks free will. The player can only fool the superbeing if they do have free will. So in effect, the superbeing says to the player “You have no free will” and the player responds “I do have free will”. One of them is correct, and the other is not. This isn't a paradox, but simply a question of whether people have free will or not. It is a question which has not yet been solved, with both sides making strong logical arguments, but there is an answer and so one of the two beings is wrong. There is no paradox.

Solution 2: In the first solution, I assumed that classical physics applied. However we know that classical physics is not complete, because quantum effects have been studied and measured experimentally for over a century now. So what does quantum mechanics say about Newcomb's paradox? Quantum mechanics says that the superbeing can create a virtual state in which box B contains both the million dollars and nothing at the same time (for those who understand the thought experiment of Schrodinger's cat, this is the same effect). Then the player makes their choice, and the contents of box B change from a virtual superposition of two prizes to a single real prize, based on the player's choice. From the player's perspective, the superbeing has successfully predicted their choice, but in truth the superbeing made two predictions and allowed nature to discard the wrong one. (Which coincidentally is how many mentalist tricks work :) )

So that is Newcomb's paradox in a nutshell. It is fun to think about, but in my own humble opinion it isn't truly a paradox because there are these two simple solutions which involve nothing more than properly defining the problem itself.