When Can You Debias an LLM Judge? Identifiability Limits, a Test, and Designs for Top-k Ranking

Large language models (LLMs) are increasingly used as cheap, scalable judges that compare candidate outputs pairwise. Because such judges prefer verbose or well-formatted answers, the natural fix is to add bias covariates to a Bradley--Terry model and estimate the bias away. We show this cannot work as advertised: the quality/bias split is not identified by pairwise comparisons, and the failure is exact -- across 48 real judge-pools the profile likelihood over the coefficient is flat to 0.0000 nats, and scaling the comparisons 26× buys none. A ``debiased'' score is selected by the prior, not recovered from data. Our contribution is accordingly not a better estimator but a characterization of when prior-based correction is justified, plus designs that supply the missing information when it is not. The assumption the prior encodes -- quality is a priori uncorrelated with the covariate -- pays only while corr(θ,x) stays below a crossing point (configuration-dependent, 0.22--0.60), which is what makes the same model help on LLMBar and hurt on SummEval and Nectar. We give two escapes: a trusted-anchor gate that decides per (judge, covariate, task) (no false enables in 6,000 decisions at K≥10 anchors, a rate our sample bounds at ≤6%), and a paired rendering design. Across fifteen real LLM judges bias is heterogeneous and capability-dependent: correction improves topk recall by 0.20--0.32 on five biased-but-competent cheap judges and is a no-op on frontier ones (Spearman ρ=-0.84 between competence and gain over the 14 competent judges, p<10-3), concentrating the benefit where at-scale evaluation happens.