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Pairwise comparison data arises in many domains, including tournamentrankings, web search, and preference elicitation. Given noisy comparisons of afixed subset of pairs of items, we study the problem of estimating theunderlying comparison probabilities under the assumption of strong stochastictransitivity (SST). We also consider the noisy sorting subclass of the SSTmodel. We show that when the assignment of items to the topology is arbitrary,these permutation-based models, unlike their parametric counterparts, do notadmit consistent estimation for most comparison topologies used in practice. Wethen demonstrate that consistent estimation is possible when the assignment ofitems to the topology is randomized, thus establishing a dichotomy betweenworst-case and average-case designs. We propose two estimators in theaverage-case setting and analyze their risk, showing that it depends on thecomparison topology only through the degree sequence of the topology. The ratesachieved by these estimators are shown to be optimal for a large class ofgraphs. Our results are corroborated by simulations on multiple comparisontopologies.

Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons