Generalized Bradley-Terry Models for Score Estimation from Paired Comparisons

TL;DR

This paper introduces a broad class of probabilistic models called generalized Bradley-Terry models that can handle various types of paired comparison data, ensuring convexity, stability, and efficient computation for score estimation.

Contribution

The paper develops a unified framework for paired comparison models within the exponential family, guaranteeing convexity, monotonicity, and Lipschitz stability of the MAP estimates.

Findings

GBT models are strictly convex and include classical models.

MAP scores are monotonic with respect to comparisons.

Scores are Lipschitz-resilient to new comparisons.

Abstract

Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory or defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.