Testing for the Bradley-Terry-Luce Model

The Bradley-Terry-Luce (BTL) model is one of the most widely used models for ranking a set of items given data about pairwise comparisons among them. While several studies in the literature have attempted to empirically test how accurately a BTL model can model some given pairwise comparison data, this work aims to develop a formal, computationally efficient hypothesis test to determine whether the BTL model accurately represents the data. Specifically, we first propose such a formal hypothesis test, establish an upper bound on the critical radius of the proposed test, and then provide a complementary lower bound on the critical radius. Our bounds prove the minimax optimality of the scaling of the critical radius with respect to the number of items (up to constant factors). Finally, we also take the first step towards characterizing the stability of rankings under the BTL model when there is a small model mismatch.