  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. This condition is classical in the social sciences literature (see
Fishburn(1973)) and has been studied in the statistics literature in recent
work (see Chatterjee (2015) and Shah et al. (2016)). In this paper, we
investigate in detail a natural estimator which bypasses the need to search
over all possible latent permutations and hence is computationally tractable,
and derive asymptotic risk bounds for our estimator. In addition, we prove an
automatic adaptation property of our estimator for several sub classes of our
parameter space which are of natural interest. These sub classes include
generalizations of the popular Bradley Terry Model in the Tournament case, the
Beta model and Stochastic Block Model in the Graph case, and Holder smooth
matrices in the tournament and graph settings.
