Estimation in Tournaments and Graphs Under Monotonicity Constraints

We consider the problem of estimating the probability matrix governing a tournament or linkage in graphs from incomplete observations under the assumption that the probability matrix satisfies natural monotonicity constraints after being permuted in both rows and columns by some latent permutation. We propose a natural estimator which bypasses the need to search over all possible latent permutations and hence is computationally tractable. We then derive asymptotic risk bounds for our estimator. Pertinently, we demonstrate an automatic adaptation property of our estimator for several sub classes of our parameter space which are of natural interest, including generalizations of the popular Bradley-Terry model in the tournament case, the $\beta $ model and stochastic block model in the graph case, and Holder continuous matrices in the tournament and graph settings.