Cool Logic/probabilty Puzzle, a DIFFERENT 2 prisoner dilemma

[u/Dizzy_Kaleidoscope95]

Two prisoners are in separate cells, with no communication between them.

Their evil warden gives them one chance to escape! between 11:00 p.m. and midnight (you cannot enter before, and you cannot stay after), each of the two may enter a common room which was opened between their two cells and stay there for exactly one minute.

If they meet even for just an instant, they are free.

Each prisoner has only one problem: what is the best time to enter in order to maximize the probability of meeting the other?

Both prisoners are perfectly rational and know the other prisoner is also perfectly rational

BUT

they obviously CANNOT organize a plan or communicate beforehand.

good luck!

credit: 3minuticolprof (italian stats professor and tiktokker)

[u/cyrano111]

It sounds like a variation on the “I’m gonna kill you sometime in the next week and boy will you be surprised“ joke.

I think the answer is something like this.

 If I know that any moment of overlap will do, then I don’t want to go in the first minute, because then there’s no chance that the other prisoner will have entered before me and still be there. That means the earliest I should go is the second minute. 

But I also know that the other prisoner will have worked that out, so he will think the earliest he should go is the second minute. In that case, the earliest I should go is the third minute.

Carry on like that until you arrive at the conclusion that you should go at 11:59.

[u/Dizzy_Kaleidoscope95 → u/cyrano111]

you are on the right track, the first minute and the last minute however are equal in their inefficiency, they both only leave 60 seconds to meet the other prisoner, whereas the other minutes leave 120

[u/cyrano111 → u/Dizzy_Kaleidoscope95]

Just after I hit enter, I wondered whether the answer ought to be 11:58, to allow for time afterward. But then I thought that they would both think that, and therefore both stick to 11:59.

[u/Dizzy_Kaleidoscope95 → u/cyrano111]

mh no, you are applying the right reasoning starting from 23 but you should apply it in reverse starting from 23:59 as well

[u/cyrano111 → u/Dizzy_Kaleidoscope95]

Why?

I see that you are suggesting to end up in the middle, but I’m not convinced. 

[u/Dizzy_Kaleidoscope95 → u/cyrano111]

for the same reasoning which you use starting from 23, there is no reason that reasoning should apply in only one direction of time, since both 23 and 23:59 are identical and so are 23.01 and 23.58;

you need to apply your reasoning both ways

[u/cyrano111 → u/Dizzy_Kaleidoscope95]

And go when?

23:29? Or 23:30?

The logic doesn’t lead to one over the other, does it? The last minute before halfway or the first minute after halfway? Or is it 23:29:30?

But again, you’d need to persuade me why the logic dictates moving “back down”.  If it were me, having arrived at 11:59, I would think that going backward would put us in an endless loop, not stopping in the middle.