Distinct Extreme Scores in Random Round-Robin Tournaments

Abstract

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 $k(n)\to\infty$ as $n\to\infty$ and $\frac{k(n)^2\sqrt{\log(n/k(n))}}{\sqrt n}\to 0,$ then, with probability tending to one, the largest $k(n)$ scores are all distinct. In particular, this holds whenever $k(n)=o\!\Bigl(\bigl(n/\log n\bigr)^{1/4}\Bigr).$ 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 Erd\H{o}s--R\'enyi 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.