You’re lost in a forest. What’s the best way to get out?
The great macroeconomist Bob Lucas once asked me this question, and I had no answer for him. I still don’t.
The assumption is that you know the size and shape of the forest, but you don’t know where you are or which way you’re facing. And the forest is so dense that you can never see any significant distance in front of you. What path should you follow?
The answer depends on your goal. Are you trying to minimize your escape time in the average scenario (the average being taken over all possible starting positions)? Or are you trying to minimize your escape time in the worst-case scenario? Either way, I don’t know the answer.
The easiest case might be a forest in the shape of a disk, say of radius 1. Then if you walk a straight line in a random direction, the distance to the boundary is 4/3 on average and 2 in the worst possible case. (The worst case, of course, is when you start right on the edge and set off in exactly the wrong direction.)
I expect you might do better to spiral, though. That way, if you start off anywhere near an edge, you’re sure to get out in a hurry. I suspect (but do not know) that with the right spiral, you could beat that 4/3 average, though the worst case might be pretty bad. How tight should the spiral be? I dunno.
If you’re in a long narrow rectangular forest, a spiral seems like an even better bet, and this time it looks pretty good even in the worst case. But maybe something else is better.
It might be reasonably straightforward to set this up as a problem in the calculus of variations, but I haven’t done that. Nor have I Googled around to find out if this is a well-known problem (perhaps with a well-known solution). All thoughts are welcome thoughts.