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We study methods for aggregating pairwise comparison data in order toestimate outcome probabilities for future comparisons among a collection of nitems. Working within a flexible framework that imposes only a form of strongstochastic transitivity (SST), we introduce an adaptivity index defined by theindifference sets of the pairwise comparison probabilities. In addition tomeasuring the usual worst-case risk of an estimator, this adaptivity index alsocaptures the extent to which the estimator adapts to instance-specificdifficulty relative to an oracle estimator. We prove three main results thatinvolve this adaptivity index and different algorithms. First, we propose athree-step estimator termed Count-Randomize-Least squares (CRL), and show thatit has adaptivity index upper bounded as $\sqrt{n}$ up to logarithmic factors.We then show that that conditional on the hardness of planted clique, nocomputationally efficient estimator can achieve an adaptivity index smallerthan $\sqrt{n}$. Second, we show that a regularized least squares estimator canachieve a poly-logarithmic adaptivity index, thereby demonstrating a$\sqrt{n}$-gap between optimal and computationally achievable adaptivity.Finally, we prove that the standard least squares estimator, which is known tobe optimally adaptive in several closely related problems, fails to adapt inthe context of estimating pairwise probabilities.

Feeling the Bern: Adaptive Estimators for Bernoulli Probabilities of Pairwise Comparisons