How to feel settled: an optimal stopping problem

Imagine you’re a beaver. You live on a river, and you’re looking for a good place to build your beaver dam.

You go swimming down the river and you pass a bunch of spots that don’t seem like they’d work well. Soon you come across an area with a nice cluster of boulders that form a small, protected eddy. This might do. But it would be a little small, you think. Not much room to expand. You keep swimming.

Soon another potential dam location catches your eye. It’s an area where a group of large boulders lie underwater churning a deep, concentrated stream. There are already fallen logs that have caught and formed a partial dam. But here there are huge willows overhanging the whole area. Not much sunlight would hit your home, if you built it here. You hang around. But something in you is restless. You’re still not sure about this spot. Of all the places on this vast river for you to make your forever home, is this the one? Is this the best you can do? Finally you decide it isn’t working, and you leave.

You continue on your journey in much the same way: swimming, finding a potential location, hanging out for a while, eventually moving on. The months pass. Until eventually you find yourself in the Gulf of Mexico, without a home. A hillbilly human throws a rock at you and you die. That sucks.

But it COULD have gone another way. What if you hadn’t cared so much about sunlight? If you had been the kind of beaver who could accept a shady dam, you’d be alive and well, living a flourishing beaver life. Instead you’re floating in the Gulf of Mexico, killt by a hillbilly’s rock.

Fine, you say, and if the horse had a horn it’d be a unicorn. So what. You were in fact this kind of beaver, and you couldn’t have simply chosen to be another kind of beaver.

I hear ya.

But every thought that passes through your mind, and every scene of life you experience, will change you. If you believe it would be better for you to live the shady dam life than the Gulf death, then there are experiences you can put yourself through by which you can gradually grow into the kind of beaver who’s skilled at enjoying the life he has.

One of them is this thought: the explore vs exploit problem — should you continue swimming down the river, losing time but chancing after a better home, or should you settle on this one, which is pretty good but not perfect? If you search too long, you’ll wind up killt by a hillbilly in the Gulf of Mexico. If you search too short, you’ll wind up with an unlivable dam.

This is an example of an optimal stopping problem. And it turns out there’s an optimal solution in terms of expected utility. First, decide how long you want to spend searching. If you’re floating down the Mississippi, that amount of time might be however long it takes to reach the Gulf. Then, here’s what you do: spend the first 37% of your search period evaluating options, but do not select one. Just keep track of the best one you’ve seen yet. After the 37% point, as soon as you come upon one option that’s as good as the best you’ve seen, select it and commit on the spot.

You can’t be certain this method will get you the very best option. There’s no way to guarantee that. But you can be confident that you did the very best you could. And that knowledge should help you to be the kind of beaver who is skilled at enjoying what he has.

For more on this, read Algorithms to Live By.


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