Extreme Score Distributions in Countable-Outcome Round-Robin Tournaments of Equally Strong Players

TL;DR

This paper analyzes the asymptotic distribution of extreme scores in large, equally strong round-robin tournaments with countable outcome scores, providing limiting distributions and convergence rates.

Contribution

It introduces a general model for such tournaments, derives asymptotic distributions of extreme scores, and addresses computational intractability of exact distributions.

Findings

Derived limiting distributions for maximum and other extreme scores.

Established rates of convergence as number of players increases.

Provided asymptotic results applicable to large tournaments.

Abstract

We consider a general class of round-robin tournament models of equally strong players. In these models, each of the $n$ players competes against every other player exactly once. For each match between two players, the outcome is a value from a countable subset of the unit interval, and the scores of the two players in a match sum to one. The final score of each player is defined as the sum of the scores obtained in matches against all other players. We study the distribution of extreme scores, including the maximum, second maximum, and lower-order extremes. Since the exact distribution is computationally intractable even for small values of $n$, we derive asymptotic results as the number of players $n$ tends to infinity, including limiting distributions, and rates of convergence.