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 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.
From: Yaakov Malinovsky [view email]
[v1]
Fri, 6 Mar 2026 21:07:15 UTC (9 KB)
[v2]
Sat, 30 May 2026 13:24:59 UTC (11 KB)
[v3]
Thu, 25 Jun 2026 20:45:04 UTC (12 KB)
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