Bradley-Terry model in random environment : does the best always win?

Raphael Chetrite (1) , Roland Diel (1) , Matthieu Lerasle (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

We consider in this article a championship where the matches follow a Bradley-Terry model in random environment. More precisely the strengths of the players are random and we study the influence of their distributions on the probability that the best player wins. First we prove that under mild assumptions, mainly on their moments, if the strengths are unbounded, the best player wins the tournament with probability tending to 1 when the number of players grows to infinity. We also exhibit a sufficient convexity condition to obtain the same result when the strengths are bounded. When this last condition fails, we evaluate the number of players who can be champion and the minimal strength for an additional player to win. The proofs are based on concentration inequalities and the study of extreme values of sequences of i.i.d. random variables.

Mots clés: Bradley-Terry model; random environment; paired comparison; complex systems

Domaines: Probabilités [math.PR]

Raphael Chetrite, Roland Diel, Matthieu Lerasle. Bradley-Terry model in random environment : does the best always win?. 2015. ⟨hal-01204517v1⟩