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

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${{n \choose 2}}$games, that is, 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.