The number of potential winners in bradley.terry model in random environment

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.