  Pairwise comparison data arises in many domains, including tournament
rankings, web search, and preference elicitation. Given noisy comparisons of a
fixed subset of pairs of items, we study the problem of estimating the
underlying comparison probabilities under the assumption of strong stochastic
transitivity (SST). We also consider the noisy sorting subclass of the SST
model. We show that when the assignment of items to the topology is arbitrary,
these permutation-based models, unlike their parametric counterparts, do not
admit consistent estimation for most comparison topologies used in practice. We
then demonstrate that consistent estimation is possible when the assignment of
items to the topology is randomized, thus establishing a dichotomy between
worst-case and average-case designs. We propose two estimators in the
average-case setting and analyze their risk, showing that it depends on the
comparison topology only through the degree sequence of the topology. The rates
achieved by these estimators are shown to be optimal for a large class of
graphs. Our results are corroborated by simulations on multiple comparison
topologies.
