Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons
This addresses a fundamental limitation in ranking and preference elicitation by establishing a worst-case vs average-case design dichotomy, which is incremental but clarifies conditions for reliable estimation.
The paper tackles the problem of estimating underlying comparison probabilities from noisy pairwise comparisons under strong stochastic transitivity, showing that consistent estimation is impossible for arbitrary item assignments but possible with randomized assignments, with proposed estimators achieving optimal rates dependent on the topology's degree sequence.
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.