I'll bite, but how do you know it converges to 1/e if no one has figured out why? Your description sounds close to the wikipedia description of the first round

I'll bite, but how do you know it converges to 1/e if no one has figured out why? Your description sounds close to the wikipedia description of the first round of that movie (https://en.wikipedia.org/wiki/13_Tzameti https://en.wikipedia.org/wiki/13_Tzameti), so hopefully it's not just the wrong probability to analyze, but naively, the setup sounds like it would have slightly above 5/6 probabilty of survival for the first round. But some aspects of the question setup also almost remind me of the classic networking algorithm of slotted Aloha, which does yield an optimal utilization (packet survival) rate of 1/e in each round (for an altered question setup).