I heard something in a podcast with which my intuition does not agree. I will paraphrase, because I don't remember the exact wording, but here's the idea:
Four people are blindfolded, back to back, such that the first one faces North, the second one faces East, the third one faces South and the fourth one faces West. They are on some two-dimensional finite plane with mountains (with some peeks and some valleys). Their goal is to find the highest point on the plane (the highest mountain).
Let their algorithm for finding the peak be the following: Everyone takes a step in the direction they are facing and say whether they went upward or downward. They then all take a step in the direction of one of the people who said he went upward (if there are more than one, they agree on one of them, say randomly). The algorithm ends when all of them say that they are taking a step downward. According to the speaker, they are then at the nearest peak. I agree with that statement, with the caveat that it might not be the nearest in terms of Euclidean distance, but, at least, it's some kind of local peak that is not necessarily the highest peak of them all.
Now, the speaker says that if we introduce some kind of noise (or randomness?) in the sense that the four people will, with a probability p (I assume it should be smaller than 50%, but the speaker did not mention that) state a wrong statement (for example, saying that they went upward, but they in fact went downward in their direction), then they will end up at the highest peak on the plane.
Now.. it is rather obvious that if the algorithm ends when all of them say they are going downward, they might end up at the highest peak, but they might not. But even if we never end the algorithm, and just say that they do not take a step in any direction if the four of them say that their direction is going downwards, it seems to me that they will, in general, spend the plurality of their time (I'm not sure of the terminology, I mean more time than around any other peak, but not necessarily more than 50%) around the highest peak, but they will go everywhere and will always end up somewhere else and will never get stuck (for an infinite amount of time) around the highest peak. I am also pretty confident that they will not spend 100% of their time around the highest peak, whatever the probability p is.
This is only my intuition, and I don't know how to try to prove my claim. I believe I can use Markov chains, but am not sure on how to do so.
I know the question is not super-well formulated (both due to my lack of math knowledge and English), but, if someone can offer a better phrasing of the problem, or some incomplete solution or prove me wrong, I will be very grateful.
Thank you in advance
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Your intuition is accurate. To clarify some points, lets use the language of Markov Chains.
Under the initial ruleset, it it impossible to lose altitude. As such, many states are inaccessible (can never be reached, by any series of steps) from others. The peaks themselves are recurrent, but all other states are transient.
On the contrary, once any amount of randomness is added, all positions are accessible from any other. If you're lucky enough, you could even end up taking the shortest possible path from the highest peak to the deepest valley. If every position is accessible, then they all must be recurrent, meaning that they all should happen (eventually) given enough time. This means the highest mountain will eventually be reached, but so will the lowest valley. It further means they won't spend 100% of their time near that highest peak, because they spend time visiting everywhere else too.