On the Size of the Largest Distinct Extreme Score Set in Random Round-Robin Tournaments

We consider a general round-robin tournament model with equally strong players, where $X_{ij}$ denotes the score of player $i$ against player $j$. We assume that $X_{ij}$ takes values in a countable subset of $[0,1]$ and satisfies $X_{ij}+X_{ji}=1$. We prove that if $k(n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ and $\frac{k(n{)}^2\log \text{\hspace{0.17em}⁣}(n\mathrm{/}k(n))}{\sqrt{n}}\to 0,$ then, with probability tending to one, the largest $k(n)$ scores are all distinct. In particular, this holds whenever $k(n)=o\text{\hspace{0.17em}⁣}\left(\right(n\mathrm{/}\log n{)}^{1\mathrm{/}4})\mathrm{.}$ By symmetry, the same conclusion also holds for the lowest $k(n)$ scores. The obtained scale coincides with the one arising in classical problems on distinct extreme degrees in Erdős-Rényi random graphs, despite the fundamentally different dependence structure. This suggests that distinctness of extreme values may persist under broad classes of models exhibiting weak dependence.