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

TL;DR

This paper investigates the size of the largest set of distinct extreme scores in random round-robin tournaments with equally strong players, establishing conditions under which these scores are almost surely all distinct.

Contribution

It provides probabilistic bounds on the size of the largest set of distinct extreme scores in such tournaments, extending understanding of score distributions.

Findings

Largest $k(n)$ scores are all distinct with high probability under specified conditions.

Symmetry implies the lowest $k(n)$ scores are also all distinct.

Conditions involve growth rates of $k(n)$ relative to $n$ and logarithmic factors.

Abstract

We consider a general round-robin tournament model with equally strong players in which $X_{ij}$ 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\log\!\bigl(n/k(n)\bigr)}{\sqrt{n}}\to 0, $ then with probability tending to one, the largest $k(n)$ scores are all distinct. By symmetry, the same conclusion holds for the lowest $k(n)$ scores.