The existence of maximum likelihood estimates in the Bradley-Terry model and its extensions

TL;DR

This paper investigates conditions under which maximum likelihood estimates in the Bradley-Terry model and its extensions become degenerate (0 or 1), providing algorithms to identify such cases and methods for estimating remaining probabilities.

Contribution

It introduces algorithms to detect degenerate estimates in the Bradley-Terry model and its extensions, improving the understanding of estimation issues in paired comparison models.

Findings

Identifies data configurations leading to degenerate MLEs.

Provides algorithms to detect outcomes with probability 0 or 1.

Suggests methods to estimate and summarize remaining probabilities.

Abstract

In the Bradley-Terry model for paired comparisons, and its extensions to include order effects and ties, the maximum likelihood estimates of probabilities of certain outcomes can be 0 or 1 under certain data configurations. This poses problems for standard estimation methods. In this paper, we give algorithms for identifying the outcomes with estimated probability 0 or 1, and indicate how the remaining probabilities may be estimated and summarized.