To stay discovered: On tournament mean score sequences and the Bradley--Terry model
David Aldous, Brett Kolesnik
TL;DR
This work analyzes which mean-score sequences can arise in random tournaments and connects them to convex order and the Bradley–Terry (BT) model. It provides two probabilistic, constructive proofs of Moon's theorem—one via a football-game construction and one via random total orders and doubly stochastic matrices—and proves a BT-closure result showing that the BT-obtained mean scores are dense in the full feasible set. It also clarifies the max-entropy interpretation of BT as the entropy-maximizing solution under row-sum constraints and discusses strong stochastic transitivity and related properties. By situating these results within convex order and Strassen representations, the paper clarifies how mean tournament performance can be modeled and approximated, with implications for joint-distribution constructions and broader probabilistic order theory.
Abstract
On being told that a piece of work he thought was his discovery had duplicated an earlier mathematician's work, Larry Shepp once replied "Yes, but when {\em I} discovered it, it {\em stayed} discovered". In this spirit we give discussion and probabilistic proofs of two related known results (Moon 1963, Joe 1988) on random tournaments which seem surprisingly unknown to modern probabilists. In particular our proof of Moon's theorem on mean score sequences seems more constructive than previous proofs. This provides a comparatively concrete introduction to a longstanding mystery, the lack of a canonical construction for a joint distribution in the representation theorem for convex order.