## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 3 (2017), 1372-1394.

### The number of potential winners in Bradley–Terry model in random environment

Raphael Chetrite, Roland Diel, and Matthieu Lerasle

#### 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 moment and convexity conditions, the asymptotic probability that the best player wins is 1. The convexity condition is natural when the distribution of strengths is unbounded and, in the bounded case, when this convexity condition fails the number of potential winners grows at a rate depending on the tail of the distribution. We also study the minimal strength required for an additional player to win in this last case.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1372-1394.

**Dates**

Received: October 2015

Revised: July 2016

First available in Project Euclid: 19 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1500451225

**Digital Object Identifier**

doi:10.1214/16-AAP1231

**Mathematical Reviews number (MathSciNet)**

MR3678473

**Zentralblatt MATH identifier**

1372.60138

**Subjects**

Primary: 60K37: Processes in random environments 60G70: Extreme value theory; extremal processes 60K40: Other physical applications of random processes

**Keywords**

Bradley–Terry model random environment paired comparisons extreme values

#### Citation

Chetrite, Raphael; Diel, Roland; Lerasle, Matthieu. The number of potential winners in Bradley–Terry model in random environment. Ann. Appl. Probab. 27 (2017), no. 3, 1372--1394. doi:10.1214/16-AAP1231. https://projecteuclid.org/euclid.aoap/1500451225