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Abstract:
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The Bradley-Terry model is designed to study tournament matrices $\Theta$, where each team has a parameter $p_i$ quantifying "team strength", and $\Theta_{ij}=\mathbb{P}$(team $i$ beats team $j$)=\frac{p_i}{p_i+p_j}$. The non-parametric Bradley-Terry model generalizes this by positing only suitable monotonicity restrictions on the matrix $\Theta$. Estimating $\Theta$ in this set up using Least Squares is computationally prohibitive, because we typically do not know the correct ranking of the team strengths. We propose a natural computationally efficient estimator for $\Theta$, which uses a plugin estimate for the underlying correct ranking, and study its global worst case risk, as well as its adaptive properties on some special sub cases: (i) a block tournament matrix, (ii) the usual Bradley-Terry model, and (iii) smooth (Lipschitz/Hölder's continuous) matrices.
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