# Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence

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Nihar Shah,
Sivaraman Balakrishnan,
Joseph Bradley,
Abhay Parekh,
Kannan Ramchandran,
Martin Wainwright
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Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:856-865, 2015.

#### Abstract

Consider the problem of identifying the underlying qualities of a set of items based on measuring noisy comparisons between pairs of items. The Bradley-Terry-Luce (BTL) and Thurstone models are the most widely used parametric models for such pairwise comparison data. Working within a standard minimax framework, this paper provides sharp upper and lower bounds on the optimal error in estimating the underlying qualities under the BTL and the Thurstone models. These bounds are are topology-aware, meaning that they change qualitatively depending on the comparison graph induced by the subset of pairs being compared. Thus, in settings where the subset of pairs may be chosen, our results provide some principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors. We use this result to investigate the relative merits of cardinal and ordinal measurement schemes.

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