Ah, I never thought about this being a secretary problem.
Well, initially I used it as an analogy for evolution and didn’t think too much about memorising/backtracking.
Oh wait, the mountaineer has memory about each peak he saw then he should go back to one of the high peaks he encountered before (assuming the flood hasn’t moped the floor yet, which is a given since he is still exploring), there is probably no irrecoverable rejections here like in secretary problem.
The second choice is a strange one. I think the entire group taking the best chance on one peak ALSO maximises the expected number of survivals, together with maximising each individual’s chance of survival.
But it still seems that “a higher chance that someone survives” is something that we want to take into the utility calculation when humanity wants to make choices in face of a catastrophes.
For example, if a coming disaster gives us two choices
(a): 50% chance that humans will go extinct, 50% chance nothing happens.
(b): 90% chance that 80% of humans will die.
The number of expected deaths of (b) significantly exceeds the one of (a), and (a) expects a greater number of humans surviving. But I guess many will agree that (b) is the better option to choose.
Yes, those two pieces can change the situation dramatically (and I have tried writing another parable including them, but found it a bit difficult for me)
I’m pondering about what is the best strategy with communication. Initially I thought I can spread them out and each mountaineer knows the location/height of other mountaineers in a given radius (significantly larger than the visibility in the fog) and add that information into their “move towards the greatest height” algorithm. Which might work, but I cannot vigorously show how useful that will be.
Regardless, I think evolution can’t get much better than the third scenario, it doesn’t seem to backtrack and most likely doesn’t communicate.
There is also that my analogy fails to consider that the “environment” changes overtime, so the “mountain landscape” will not stay the same when you come back to a place after leaving it. This probably prevents backtracking, but doesn’t change the outcome that you’d most likely be stuck on a hilltop that isn’t the optimal.