A group of people all randomly pick another member in the group. How many people are not picked by anybody? {see details}?

Anybody else like my use of curly brackets in the title? I’m trying to be different.

There is a group of 100 people at a party, numbered 1–100. Each person independently picks a random integer in the same range, excluding their own number. If a person has his assigned number picked by at least one person, then he gets cake. On average, how many people do not have their number picked by anybody?

Because of the well known birthday paradox, it likely that at least two people out of the group will pick the same number, so it is very likely that at least one person will walk away with no cake.

A general solution to this problem would be some formula such that given X people at the party, it gives Y average number of people without cake.

I am interested in this problem because of the 80–20 rule. There is guaranteed to be a number N such that N% of the people received (100-N)% of the votes, and I wonder if that number will be 80 or 20.

This problem at first reminds me of the birthday paradox, but it is different in that the number of numbers that each person could potentially pick is not a constant 365, but varies with the size of the group.

I hope that I have explained this problem well enough. Feel free to ask any questions.