Pair Difficulty Matters: Rethinking Pairwise LLM-as-a-Judge Evaluation and Consistency

Large Language Model judges are widely used to rank texts and text-generating systems through pairwise comparison, and their reliability is typically assessed via three proxies: position bias, transitivity, and pairwise agreement (self- or human-labeled). Because these proxies drive judge selection and benchmarking, a substantial literature reporting that judges perform poorly on them risks steering practitioners away from otherwise capable evaluators. We argue this assessment is misleading. Under the Bradley–Terry geometry underlying pairwise aggregation, each proxy is dominated by close-rank-gap pairs, where inconsistency is information-theoretically expected and individual verdicts contribute little to the aggregate ranking; far-gap pairs carry the ranking signal but barely move the proxies. We formalize this argument and validate it in a controlled simulation and on two human-rated corpora: the proxies correlate only weakly with ranking accuracy against gold, and their predictive component concentrates in the far-gap regime. Judges should therefore be assessed on rank-gap-conditional metrics, ideally against human rankings. Code at https://github.com/brunobrocai/PairDifficulty.