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 in which Xᵢⱼ denotes the score of player i against player j. We assume that Xᵢⱼ takes values in a countable subset of $[0,1]$ and satisfies Xᵢⱼ+Xⱼᵢ=1. We prove that if k(n)→∞ as n→∞ and k(n)²log\!(n/k(n))√n→ 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.