Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $p(n)$ denote the probability that there is only one vertex with the maximum out-degree. What is $\lim\_{n\rightarrow\infty}p(n)$?


I would be surprised if this statement hasn't been written and proved anywhere, but I can't find it. Does anyone know of a reference, or a result that implies it?



Unique maximum degree in tournament
I can't find a published proof of this known result, but here is a close miss.


In [this paper](https://core.ac.uk/reader/82426260), page 256, is a short proof that a random undirected graph has a unique vertex of maximum degree almost surely. If you replace "graph" by "tournament" and "degree" by "out-degree", the exact same proof works for tournaments.


The reason this works is that the degree of a given vertex in a random graph, or out-degree in a random tournament, have the same binomial distribution Bin$(n-1,\frac12)$. Moreover, the degrees/out-degrees of different vertices are almost independent.


