I am amazed and delighted by the many excellent responses to my call for arguments about religion. These will be very helpful to me as I prepare for my debate with Dinesh D’Souza. Keep them coming!
Commenters also had a lot to say about the puzzle of the absent-minded driver. It seems to me that some of the analyses falter by being less than crystal clear about their assumptions: How much can the driver remember (e.g., if he updates his strategy at the first intersection and then arrives at the second, does he remember his original strategy or his updated strategy?), how much he can commit to (e.g. if he updates his strategy at the first intersection, can he commit to sticking to the new strategy at the second? can he commit to the method of updating he’ll use at the second?), how much he can anticipate (e.g. what does he believe about his future updates?), how smart he is (can he use his knowledge of his current strategy to figure out whether he’s already updated and hence what intersection he’s at?) and how sneaky he is (e.g. might he purposely adopt a bad strategy in order to trick his future self into updating to a good one?). I have what I think is a useful way of forcing puzzlers to be explicit about their assumptions, and had planned to post it on Monday, but I keep revising it, so it might be a few more days.
Well, to be crystal clear about my assumptions, it is that when Albert finds himself at an intersection, whether it be the first or the second, he has exactly the same information and makes the same inferences. Further, since he can look forward, he knows that if he goes straight and encounters another intersection, he will be in EXACTLY the same situation again. From this symmetry of ignorance he can deduce that he can do no better than flip a fair coin.
Steven,
What do you think about Rand Paul’s comments about the Civil Rights Act?
Neil: From this symmetry of ignorance he can deduce that he can do no better than flip a fair coin. I don’t see how this follows (and in fact it seems to me not to follow, but maybe I don’t understand what you’re saying). Can you post your math?
If Albert turns with a 50% probability, then when he reaches an intersection, the odds of it being the first intersection are 2/3. If he therefore adjusts his turn probability to 1/3, now his odds of making it home are 2/9. Of course, he now needs to adjust the probability of being at the first intersection, which are now 3/5. But then if he adjusts his turn probability to 2/5, he increases his odds of making it home to 6/25, or 24%.
Albert can continue to adjust his turn probability based on his odds of being at the first intersection, and if he continues indefinitely, the odds of his being at the first intersection approach the golden ratio, 61.8%, because the odds of being at the first intersection are f(n+2)/f(n+3), where f(n) is a number in the Fibonacci sequence. Unfortunately, this approach reaches a limit on the probability of Albert reaching home of 23.6%.
He would obviously be better off just sticking with a 50% turn probability.
“Can you post your math?”
I’ve done no math. I am trying to clarify assumptions. As I understand it, Albert is rational, can look forward, and can calculate, but he has no memory. Thus,
“How much can the driver remember?” Nothing.
“How much he can commit?” Nothing–he would need a memory to commit.
“How much he can anticipate?” He can anticipate that were he to reach another intersection, he will have no memory of what he did before, and will therefore be in the exactly the same boat as he is in now.
“Might he purposely adopt a bad strategy in order to trick his future self?” How can his future self be tricked? For all intents and purposes, he is his future self. His future self will either find himself at an intersection with no memory, just like he is now, or he will be dinosaur bait and it won’t matter.
Given this, how can he do anything but maximize his chances?
Neil: I have no idea what “maximize his chances” means unless you write down the function you are maximizing.
Steve: By maximizing his chances, I meant using a fair coin which gives him a 25% chance of getting home safely. He can figure that out, and he has no leverage to increase his chances under the assumptions I stated.
(I suspect you will prove me wrong next week.)
One way of committing is this: Suppose before Albert leaves his office, he can choose one and only one coin to flip. Clearly he will choose the fair coin, just in case he starts playing these mind games with himself when he finds himself at an intersection with no memory, which Albert knows will inevitably happen. If there would be any advantage to changing the strategy in the future, Albert would take a selection of coins, but he knows that can only hurt him, not help him so he will take only the fair coin.