This article was suggested by reader JH who saw it discussed at Massimo Pigliucci’s site.
When he was at the Lawrence Livermore Laboratory, William Newcomb devised the following puzzle, which some say is a paradox.
You will play a game against an evil entity that can perfectly predict your future actions. Just for the sake of giving the Evil One a name, let’s call him “Bill”. Bill is evil because he likes to taunt you with the possibility of making money, only to jerk that chance away from you at the last moment.
Here’s what Bill does, as traditionally described. Later, we add clarifications and twists. Bill puts $1,000 in a clear box, and then also presents you with an opaque box inside of which might contain $1 million. You will be allowed to take both boxes or just the opaque one. To clarify: you may not take the $1,000 without also taking the opaque box. Obviously, you may keep the contents of whatever box or boxes you take.
Now for the evil part. If Bill predicts—and remember, he’s never wrong—that you will take just the opaque box, he will leave it full of money. But if he predicts that you will greedily take both boxes, then he will put nada in the opaque box, which, after mandatory donations to our betters in Washington, nets you only about $550.
Why Bill would ever want to play this strange game is never mentioned. Anyway, the easiest way to short his pockets by a cool million is to decide now and forevermore to take just the opaque box. Because he’s never wrong, Bill will figure you’d do just this, and thus you’d walk away to find new messages on your answering machine from your brother-in-law asking if he could drop by for a chat about this new fishing boat he’s had his eye on.
By the premises of the game, there is no way you can out-think old Bill and collect the million and the thousand simultaneously. You cannot, for example, declare to the world, “I’m only taking the opaque box!” and then, at the last moment, grab both, because old Bill will have anticipated this. Nor can you attempt a crash course in hypnosis to convince yourself that you’re only taking the opaque box, only to be awakened to discover yourself picking up both. Bill would have predicted this perfectly, too.
Not much of a paradox, really. (Though there’s always someone who thinks he’s discovered a way to cheat Bill). Using just brain power and no external devices, the best you can do is a million—which would be good enough for anybody but a politician. So, in honor of our money-thirsty betters, here’s what they can do when next confronted by Bill and his boxes.
When Bill lays out his choices, take out a coin and flip it. If it comes up Heads, take both boxes; tails, take just the opaque one. As long as you are not the sort of politician who would cheat, then your actions cannot be predicted ahead of time by Bill. Since there is only a 50% chance of you taking both boxes, this is what Bill must predict.
He must, of course, come to a decision and not continuously fly between the horns of the dilemma. How Bill does this, how he, that is, comes to a definite prediction is his business, but come to one he must if he is to play his weird game.
There are problems with this. Coin flips are only “random” because it is difficult to know what the initial conditions of the flip are. But if you did know them, then the coin flip is perfectly predictable. And since it’s you who will flip the coin, and old Bill has the lock on your mental actions, we could say that he knows how you will flip the coin. Which, of course, means he knows if you’ll take one box or two.
Regardless whether this is so—you can always have a pal flip the coin, removing Bill’s prognosticative abilities, because the rules do not say Bill can predict anybody’s actions, just yours—predicating your actions on a coin flip can actually be worse for you than just picking the opaque box, because you run the chance of ending up with only $1,000. This happens when the coin lands Heads and this is what Bill predicted. The chance of this—assuming Bill follows the probabilities—is 0.5 x 0.5 = 0.25.
The chance Bill predicts Heads and you Tails is the same: meaning you will decide to take just the opaque box, but you’ll come away broken-hearted. The chance that Bill predicts Tails and you Heads is also the same, meaning there’s a 25% shot at making $1,001,000. Finally, the chance that Bill and you both go Tails is the same, meaning a 25% chance of scoring a million.
Homework: use a quantum number-maker-upper (QNMU), a device whose outcome Bill cannot prefigure, and which spits out Heads with probability p. You can either just take the opaque box, or use the QNMU to decide what to do. What strategy is best and why? If you use the QNMU, what value of p are you picking, and why?