Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence

TL;DR

This paper derives sharp minimax bounds for estimating item qualities from pairwise comparison data, revealing how the comparison graph's topology influences estimation accuracy and comparing ordinal and cardinal measurement models.

Contribution

It provides the first tight bounds on estimation error for parametric ordinal models, linking error rates to the comparison graph's Laplacian spectrum and guiding optimal pair selection.

Findings

Error bounds depend on the Laplacian spectrum of the comparison graph.

Ordinal and cardinal models have similar error rate scalings.

Guidelines for choosing comparison pairs based on graph topology.

Abstract

Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector $w^* \in \mathbb{R}^d$ that represents the "qualities" of the $d$ items being compared; this class of models includes the two most widely used parametric models--the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector $w^*$ under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via its Laplacian spectrum. Thus, in settings where the subset of pairs may be chosen, our results provide 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.