A network of secret agents, numbered from $1$ to $n$, have established the following randomized communication protocol. At a precise prearranged signal, each agent sends a message to exactly one of the other $n − 1$ agents, chosen independently and uniformly at random. For example, if $n = 3$, then agents 1 and 2 might both send their messages to agent 3, while agent 3 sends their message to agent 1.
(a)An agent is bored if no other agent sends them a message. Let $B(n)$ denote the expected number of bored agents. What is $\lim_{n\to\infty} B(n)/n$?
(b)An agent is swamped if more than one other agent sends them a message.Let S(n) denote the expected number of swamped agents. What is $\lim_{n\to\infty} S(n)/n$?
(c)Suppose each agent can accept at most one message. Thus, each swamped agent accepts one of the messages sent to them, chosen arbitrarily, and rejects the rest. (In the example scenario, exactly one message is rejected.) Let $R(n)$ denote the expected number of rejected messages. What is $\lim_{n\to\infty} R(n)/n$?
I have already computed the (a) and (b) part, which should be $\lim_{n\to\infty} B(n)/n = 1/e$ and $\lim_{n\to\infty} S(n)/n= 1 - \frac{1}{e}$.
However, I'm stuck with part (c), I try to define a indicator random variable $$R_i = \begin{cases} 1 & \text{if agent $i$'s message is rejected} \\ 0 & \text{otherwise} \end{cases}.$$ I'm stuck at computing the probability of agent $i$ message is rejected, what should be the right way of computing part (c)?
(a) looks good.
(b) In your attempt you are assuming that an agent is swamped if they receive exactly one message, which is not the definition. $$S(n)/n = P(\text{agent 1 is swamped}) = 1 - P(\text{agent 1 gets $\le 1$ messages}) = 1 - \left(\frac{n-2}{n-1}\right)^{n-1} - (n-1) \frac{1}{n-1} \left(\frac{n-2}{n-1}\right)^{n-2} \to 1 - \frac{2}{e}$$
(c) Rephrasing saulspatz's argument: There are $n$ messages. The number of accepted messages is $n-B(n)$. All other messages are rejected, so $R(n) = n - (n-B(n)) = B(n)$.