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

Yaakov Malinovsky

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.

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

Abstract

We consider a general round-robin tournament model with equally strong players in which denotes the score of player against player . We assume that takes values in a countable subset of and satisfies . We prove that if as and then with probability tending to one, the largest scores are all distinct. By symmetry, the same conclusion holds for the lowest scores.
Paper Structure (5 sections, 5 theorems, 49 equations)

This paper contains 5 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction and Problem Statement
  2. Main Result
  3. Proof of Proposition 1
  4. Proof of Proposition 2
  5. Proof of Proposition 3

Key Result

Theorem 1

If $k(n)\to\infty$ as $n\to\infty$ and then

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 1
  • ...and 4 more