What is a great riddle that has no right answer?

A group of people are standing around in a circle.

Every ten seconds, every person simultaneously shoots a random other person (with a laser tag system, of course. Nobody is actually hurt). If you're hit, you're out. Everyone is a great shot: they don't miss. Whoever they picked at random to take out is out.

Ten seconds later, once again, everyone still standing shoots someone else at random. And so it goes on until one of two things happen: either there's a last lucky person standing, or there's nobody left.

Question: what's the probability that nobody's left at the end of the game?

The answer obviously depends on the number of people we had to begin with. The probability is one thing if you start with 5 people, and another thing if you start with 50. So, being the mathematicians that we are, we're curious about the value of this probability when the initial number of people is really large – a million, a billion, etc. "As n tends to infinity", to use the common math parlance.

So, what is it? What is this limiting value of that probability? Does it tend to zero? Does it tend to 100%? Does it tend to some value in between?

The actual answer to the question as I phrased it is that there is no answer. The probability, as a function of the number of people n, does not tend to any limit. It fluctuates, without converging.

The answer to such questions is very often 0 or 1 (Zero–one law), and sometimes it's some limiting intermediate value which we may or may not be able to calculate explicitly, but it's really rare to have such naturally-defined probabilities that fluctuate for no obvious reason (it's easy to make it so that the values depend on the parity of n, for example, and then it's no surprise that they don't converge. But this isn't the case here).

Here's a graph of the probability p that nobody survives, as a function of \log(n). This is taken from the paper "The asymptotics of group Russian roulette" by van de Brug, Kager and Meester.