On Round-Robin Tournaments with a Unique Maximum Score

TL;DR

This paper proves that in large round-robin tournaments, the probability of having a unique maximum score vertex approaches 1, confirming a statement originally made without proof by Epstein in 1967.

Contribution

The paper provides a rigorous proof of Epstein's claim about the asymptotic probability of a unique maximum score in round-robin tournaments.

Findings

Probability of unique maximum score tends to 1 as n increases

Confirms Epstein's 1967 conjecture with a formal proof

Includes historical remarks and comments on the result

Abstract

Richard Arnold Epstein (1927-2016) published the first edition of "The Theory of Gambling and Statistical Logic" in 1967. He introduced some material on round-robin tournaments (complete oriented graphs) with n labeled vertices in Chapter 9; in particular, he stated, without proof, that the probability that there is a unique vertex with the maximum score tends to 1 as n tends to infinity. Our object here is to give a proof of this result along with some historical remarks and comments.