I had dinner with the great
atakdog tonight. He pointed out that my solution to the key problem is not technically correct. And to paraphrase Neil deGrasse Tyson, not technically correct is the worst kind of not correct. First my comma usage, now this! Here's the problem again:
Quote:
You leave your apartment groggily one morning, closing the door behind you. Suddenly, you are hit by a terrifying question: Do you have your keys, or are you now locked out? You stand there thinking about it for a few seconds, before deciding that yes, you probably have your keys, further estimating that 80% of the time, you have them. You also decide that there is an equal chance of your keys either being in your left pocket or your right pocket, and if they aren't in either pocket then you don't have them at all. Slowly, perversely enjoying the sweat, you slide your hand into your right pocket, and find that your keys are not there. What should you now think is the probability that your keys are in your left pocket?
My solution - in two worlds, it's in your left pocket, in two worlds, it's in your right pocket, in the last, you don't have it. You know it's not in your right pocket, so there are three possible worlds remaining. Your probability should now be 2/3.
The solution takes into account the fact that because of the new information that your right pocket is empty, 4/5 becomes 2/3. However, it fails to adjust one more probability - the chance that we were good at guessing what our probability was in the first place, given all of our possible biases in doing so.
Let's say you went through a ton of these situations and found out that over time, you were too confident you had the key with your murky initial guesses. Consistently, you thought there was a 70-80% chance of having the key when the door first closed and you first began to wonder, and you only ended up with the key half the time. Shouldn't you then, the next time it happens and you go through the same guesswork to come to 70-80%, stop yourself and say, "Wait a minute. I've been wrong about this guess before. I should guess something lower"?
Now let's go all the way back to your first key-estimating situation. You do your very best to take everything into account, and guess 80%. There's some probability you overestimate your chances in this type of scenario. There's some probability you underestimate your chances in this type of scenario. You factor all of those possible worlds in to all possible precision, and say "OK, 80%".
Now you draw the first pocket and is empty. Before we even know if we have the key or not, which is now more likely, given new information of one pocket: That you are the sort of person to underestimate the probability that you have the key in this sort of situation, or that you are the sort of person to overestimate the probability that you have the key in this sort of situation?
The best guess once the right pocket is empty isn't actually 2/3, but rather slightly lower.
It's fun reading threads in the forum and seeing the way I talk about things seep into other people's posts. It's even more fun realizing suboptimal aspects of certain strategies I employed and in broader philosophies about how to think about problems. To me, it's a good reminder of the Galfond-esque teaching point that your goal in learning about poker should not be to learn the current best wisdom of what moves to make when, but rather the best wisdom about how to think about the game, to verify what other people know, and set yourself up to move past it in the future. If you can combine confidence in-game with a skeptical approach to evaluating what you really know outside of the game, you're in a really good place. That goes for whether you're playing poker, trading stocks, being in a relationship, whatever. Find your own way to combine swagger with a merciless hunt and kill (or quarantine - don't take your irrationality too seriously, it's always going to be there) operation for your own bull****, and you're going to be in good shape.