David told me a good one last night. Here it is:
Suppose that there is a machine which can perfectly predict the future. It is never wrong. You get to play the following game with it. Two boxes are in a room and you can come into the room and open either one or both boxes. Yesterday, the machine predicted whether you would open one or both boxes, but you don’t know what it predicted you would do.
However this machine is a jerk. Last night it predicted what you would do, and snuck into the room with a bundle of cash. If the machine predicted you will open both boxes, it put no money in either box. If it predicted you would open only one box, however, it put $5000 in one box and $5000 in the other. You get to the room and must decide to open one box or two before seeing in either.
Well if you open one box and the machine predicted it, you will win some money. But, then again, if you are going to just open one box, then the perfect predictor already predicted it so there is going to be more money in the other box too. So you should just open both boxes because the money is already there. So if you are going to open one, it is rational to really open two.
Then again, the machine is a perfect predictor, and if it knew you would do this, you should only open one box. But then …