Behind Aggregate Scores: What Systematically Varying Arena-Hard's LLM Judge Reveals

Modern LLM evaluation assumes that pinning a judge to a fixed model version and decoding at temperature zero yields reproducible verdicts. We show this assumption fails as a property of how LLM-as-Judge is operationalized on cloud serving infrastructure, not of any particular model family. Across four frontier judges served via a single major enterprise cloud platform and three standard benchmarks (Arena-Hard, AlpacaEval 2, MT-Bench), identical inputs to the same temperature-zero judge, at a constant serving-reported model version, produce different verdicts across re-runs: per-item flip rates of roughly 5% on average and about 40% on the close-call items that decide leaderboard margins, with a per-judge magnitude spanning a 40x range (0.13% to nearly 10%). We introduce metrics tailored to this instability: per-item flip rate, a two-part stability profile (waver fraction and conditional intensity), and adjacency separability. For the principal judge the aggregate ranking is stable (0% top-K instability, 0% pooled winner flip); what degrades is precision: under a paired hierarchical bootstrap, roughly one-fifth to three-quarters of adjacent leaderboard positions are statistically indistinguishable, a noise floor driven mainly by finite prompt sampling rather than the judge. Across judges, leaderboards agree on the coarse ordering but diverge in the middle (Kendall's tau of 0.42-0.64 between Gemini and Sonnet judges on Arena-Hard, values sensitive to answers truncated at the generation cap). Of 15 expected head-to-head orderings we re-judge, 12 survive every re-run of the principal judge but only 8 survive every judge. Leaderboards thus report unhedged point estimates that overstate their precision. We propose a minimal, low-cost reporting protocol: several judge re-runs, published stability profiles and adjacency intervals, and results under at least two judges from different families.

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.

Large language models are often used as judges to score candidate responses, then validated with a single global metric such as correlation with reference labels. This can be misleading when the real deployment task is best-of-n selection within a prompt. In a 5,000-prompt best-of-4 benchmark from Chatbot Arena, a judge with moderate global correlation (r = 0.47) captures only 21.0% of the improvement that perfect selection would achieve over random choice. The gap arises because global agreement is driven largely by prompt-level baseline effects, while selection depends on within-prompt ranking: within-prompt correlation is only r_within = 0.27, and coarse pointwise scoring creates ties in 67% of pairwise comparisons. In a matched-pair best-of-2 audit, explicit pairwise judging recovers much of this lost signal, raising recovery from 21.1% to 61.2%. For judge-based selection, the relevant audit should report within-prompt signal, tie rates, and recovery/top-1 accuracy, not global agreement alone.