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 …
That’s not a paradox, that’s a violation of the acyclic nature of causality. You can’t use the fact that you are going to open one box to decide that you are going to open both boxes. You can’t introduce cycles into time and still expect causality to hold.
At least, that’s my opinion right now. Argue if you think I’m wrong! =)
Yeah, I had a similar intuition about what’s wrong. There is something very strange about trying to make decisions in contexts where the state of the world has already been conditioned on what you will decide. Maybe this example shows that our intuitions about rationality really aren’t fit to deal with situations with bizarre (impossible?) causality. An ability to pass information back through time apparently can make good decision making tricky. I wonder about whether there is an objectively rational thing to do.
But I think there’s still a puzzle, independent of all that: would you open one or two boxes?
I think that depends on how you define the system.
Choice
Zero-step feedback
One-step feedback
One
$5000
$0
Both
$0
$10000
If the machine placed the money based on what you are actually going to do, then you should choose to open one box. But if it placed the money based on what you were going to do and now you can switch your decision, then you should open both boxes.
I’m pretty sure it meant the first case.
Oh, that was supposed to be a table but I guess it can’t do html in the comments … each of those chunks is one row. Is this legible:
Choice——-Zero-step feedback—-One-step feedback
One—————-$5000———————-$0
Both—————-$0—————————$10000