The number of potential winners in Bradley-Terry model in random environment

TL;DR

This paper analyzes the Bradley-Terry model with random player strengths, showing how the distribution of strengths influences the likelihood of a single winner and the number of potential winners in large samples.

Contribution

It provides new theoretical results on the asymptotic number of potential winners based on the distribution of player strengths in the Bradley-Terry model.

Findings

Unbounded strengths lead to the best player winning with probability 1.

Convexity conditions ensure a single winner even with bounded strengths.

The growth rate of potential winners depends on the tail of the strength distribution.

Abstract

We consider a Bradley-Terry model in random environment where each player faces each other once. More precisely the strengths of the players are assumed to be random and we study the influence of their distributions on the asymptotic number of potential winners.First we prove that under mild assumptions, mainly on their moments, if the strengths are unbounded, the asymptotic probability that the best player wins is 1. We also exhibit a sufficient convexity condition to obtain the same result when the strengths are bounded. When this last condition fails, the number of potential winners grows at a rate depending on the tail of the distribution of strengths. We also study the minimal strength required for an additional player to win in this last case.