**Real-life motivation.** Our team has $n$ members. For the next in-team presentation session, everyone had 1 talk prepared that he or she would be able to present. Now everyone could cast $1$ vote about whose talk they would like to hear. (Everyone is modest, so no-one voted for their own talk.) Now, unfortunately, there were two talks receiving the maximum vote number, so a coin had to be tossed. - Informally speaking, I was wondering whether this dilemma situation would be less likely as $n$ grows. See formal version below.

**Formal version.** In this context, we define a directed graph $G=(V,E)$ to consist of a finite set $V$ and of a set $$E \subseteq (V\times V)\setminus\{(v,v):v\in V\}.$$
Given $v\in V$, we define the *in-degree* by $\text{deg}^-(v) = |\{a\in V:(a,v)\in E\}|$, and the out-degree is defined in a dual fashion. Let $\Delta^-(G)$ be the maximal in-degree.

In this question we only consider directed graphs $G=(V,E)$ with out-degree $\text{deg}^+(v) = 1$ for all $v\in V$ (everyone casts exactly 1 vote). Let ${\cal M}(G) = |\{v\in V: \deg^-(v) = \Delta^-(G)\}|$ be the number of vertices having maximal in-degree. (Trivially, ${\cal M}(G) \geq 1$.) Moreover, let $M_n$ be the expected value of ${\cal M}(G)$ where $G$ is an arbitrary graph with $V(G) = \{1, \ldots, n\}$ satisfying $\text{deg}^+(v) = 1$ for all $v\in V$.

Do we have $\lim\sup_{n\to\infty}M_n = 1$?