On the distribution of winners' scores in a round-robin tournament

TL;DR

This paper investigates the score distribution in a round-robin chess tournament, deriving a limit distribution for player scores as the number of players grows large, due to the complexity of exact calculations.

Contribution

It provides a novel limit distribution for player scores in large round-robin tournaments, addressing the challenge of exact distribution calculation.

Findings

Derived a limit distribution for scores as n grows large

Showed the complexity of exact score distribution calculation

Provided theoretical insights into tournament score patterns

Abstract

In a classical chess round-robin tournament, each of $n$ players wins, draws, or loses a game against each of the other $n-1$ players. A win rewards a player with 1 points, a draw with 1/2 point, and a loss with 0 points. We are interested in the distribution of the scores associated with ranks of $n$ players after ${\displaystyle {n \choose 2}}$ games, i.e. the distribution of the maximal score, second maximum, and so on. The exact distribution for a general $n$ seems impossible to obtain; we obtain a limit distribution.