Nine Judges, Two Effective Votes: Correlated Errors Undermine LLM Evaluation Panels

LLM judge panels are a standard evaluation tool, but prior work reports highly correlated panel errors: nine judges provide roughly the effective information of two independent ones, and aggregation closes only a small fraction of the gap. A natural remedy--a signal from a different evidence source, e.g., executing a test suite--produced no distinguishable change in the panel's effective-vote count at scale (-0.04, 95% CI [-0.10, +0.02]). Aggregate dependence and conditional decision utility are different questions. Elementary majority arithmetic fixes the affected set for single-ballot substitution: only decisions with a one-vote margin can change. The empirical question is whether panel error rates rise and useful substitutions concentrate there. They do: the entire accuracy gain concentrates on these pivotal queries, where it is large (+10.4 to +23.3 percentage points across three headline configurations), and is exactly zero elsewhere. We confirm the pattern across three code benchmarks and four panel sizes (a 9-judge extension and 56 dependent subsampling checks, gain +6.5 to +16.1 percentage points). On HumanEval+/MBPP+, a majority-side replacement rule raises overall accuracy from 82.44% to 85.62% while invoking the signal on 16.2% of queries; signal-only remains stronger at 87.60%. Thus population-level dependence diagnostics and margin-stratified utility are complementary, and the affected-set characterization yields a call-reduction rule for any specified single-ballot substitution policy.

LLM-as-a-Judge is now widely used to rank model outputs, train reward models, and populate public leaderboards, but its run-to-run reliability remains under-characterized. We study repeated identical evaluations on 29 tasks spanning 10 categories using two OpenAI judge models (GPT-4o-mini and GPT-4.1-mini), with 50 pairwise trials and 50 pointwise trials per question, supplemented by temperature and prompt-sensitivity ablations. Across judges, pairwise preferences flip on average 13.6% of the time, with 28% of questions exceeding a 20% flip rate and one question reaching 56%. GPT-4o-mini also exhibits a significant first-position bias (72% A-majority, p = 0.024). At the same time, mean pointwise score gaps are small (0.19--0.36 on a 10-point scale) and not statistically significant in aggregate, producing a pairwise--pointwise gap: judges frequently choose a winner even when their own scalar scores provide little evidence of a meaningful quality difference. Beyond within-judge instability, cross-judge agreement is only 76% ($κ= 0.51$), semantically equivalent prompt templates change majority outcomes in 25% of tested cases, and deterministic decoding reduces but does not eliminate inconsistency. A reliability curve analysis shows that, in our dataset, 11 repeated trials are needed for a majority vote to recover the 50-trial reference verdict with 95% probability on average, rising to 15 for high-variance questions. These findings suggest that single-trial LLM judging is often too noisy for high-stakes evaluation, and that multi-trial aggregation, position randomization, and explicit uncertainty reporting should be standard practice. Because both judges are from a single provider, cross-provider replication remains an important next step.

Multi-judge evaluation is increasingly used to assess LLMs and reward models, and the prevailing heuristic is to curate: keep the most accurate judges and discard weaker ones. We show that this heuristic can reverse when the target is not point accuracy, but calibrated probabilistic evaluation from a labeled calibration set. Holding the aggregation and calibration procedures fixed, we compare accuracy-ranked top-$k$ judge selection with using the full judge panel. Across four labeled pairwise-evaluation benchmarks spanning LLM-as-judge and reward-model settings, the calibrated full panel consistently outperforms accuracy-based selection. On RewardBench2, retaining all judges achieves negative log-likelihood (NLL) of $0.006$ versus $0.013$ under top-5 selection, halving the calibration error. This advantage persists after judge-family deduplication and against stronger same-pipeline subset search. We explain this reversal with oracle analyses showing that the optimal calibrated risk under proper scoring rules cannot increase when additional judge signals are made available, and that even below-chance judges can be useful when their biases are learnable and their signals are non-redundant. The resulting operating principle is simple: in multi-judge evaluation with labeled calibration data, do not discard weak judges by accuracy alone; keep them when they are parseable, non-redundant, and calibratable.