The lonesome king

This Riddler puzzle is about a random elimination game. Will someone remain at the end, or will everyone be eliminated?

In the first round, every subject simultaneously chooses a random other subject on the green. (It’s possible, of course, that some subjects will be chosen by more than one other subject.) Everybody chosen is eliminated. In each successive round, the subjects who are still in contention simultaneously choose a random remaining subject, and again everybody chosen is eliminated. If there is eventually exactly one subject remaining at the end of a round, he or she wins and heads straight to the castle for fêting. However, it’s also possible that everybody could be eliminated in the last round, in which case nobody wins and the king remains alone. If the kingdom has a population of 56,000 (not including the king), is it more likely that a prince or princess will be crowned or that nobody will win? How does the answer change for a kingdom of arbitrary size?

Here is my solution:
[Show Solution]