Efficient Inference for Covariate-adjusted Bradley-Terry Model with Covariate Shift

We propose a general framework for statistical inference on the overall strengths of players in pairwise comparisons, allowing for potential shifts in the covariate distribution. These covariates capture important contextual information that may impact the winning probability of each player. We measure the overall strengths of players under a target distribution through its Kullback-Leibler projection onto a class of covariate-adjusted Bradley-Terry model. Consequently, our estimands remain well-defined without requiring stringent model assumptions. We develop semiparametric efficient estimators and corresponding inferential procedures that allow for flexible estimation of the nuisance functions. When the conditional Bradley-Terry assumption holds, we propose additional estimators that do not require observing all pairwise comparisons. We demonstrate the performance of our proposed method in simulation studies and apply it to assess the alignment of large language models with human preferences in real-world applications.