Two Calls, Two Moments, and the Vote-Accuracy Curve of Repeated LLM Inference

LLM-as-a-judge panels aggregate votes from multiple models, with the expectation that diverse models yield more reliable evaluations. We develop a framework to measure the true informational value of such panels and quantify how far their reliability falls short of the independent-voting ideal. Testing a panel of 9 frontier LLMs from 7 model families on three natural language inference datasets (each with 100 human annotations per item), we find that the 9 judges effectively provide only about 2 independent votes' worth of information. Roughly three-quarters of the panel's nominal independence is lost because the models make the same mistakes on the same items. The consequences are stark: the panel's actual accuracy falls 8-22 percentage points short of what independent voting would achieve, and the best single judge matches or outperforms the full panel across all conditions. Neither adding more judges nor using smarter aggregation algorithms helps -- established methods close at most 11% of this gap, even with access to the correct answers. We quantify these findings using the Kish effective sample size (n_eff) and a Condorcet null model, and show the deficit is robust across prompt variants, temperatures, chain-of-thought reasoning, and a pairwise preference task (RewardBench). The bottleneck is correlated judges, not the aggregation algorithm, implying that scaling up panels cannot substitute for genuinely independent evaluation.

While existing benchmarks demonstrate the near-perfect performance of large language models (LLMs) on various tasks, this apparent saturation often obscures the need for rigorous evaluation of their reliability. In real-world deployment, however, achieving extremely high reliability (e.g., "five-nines" (99.999%) vs. "three-nines" (99.9%)) is fundamentally critical, as this gap results in an order-of-magnitude increase in failures, which is catastrophic in reliability-critical applications. Still, estimating such a rare failure probability with tight confidence bounds requires prohibitively large LLM inference sizes, making standard Monte Carlo evaluation infeasible under limited compute budgets. In this paper, we observe that LLM failures exhibit strong systematic patterns: across broad parameterized input spaces, a small subset of inputs disproportionately accounts for the majority of failures. Leveraging this observation, we propose to learn a sampling distribution concentrated on failure-prone inputs via the cross-entropy method (CEM). We evaluate our framework on three LLMs, Qwen2.5-Math-7B-Instruct, gpt-oss-20b-low, and Gemini 2.5 Flash Lite, across parameterized GSM8K templates and achieve up to 156.22x reduction in required inferences compared to naive uniform sampling. Our estimates reveal that models with indistinguishable accuracy on standard benchmarks can differ substantially in estimated failure rates, underscoring that reliability is a distinct and measurable axis of model quality. Our simple yet practical framework enables the evaluation of extreme reliability in LLMs, a distinct and underexplored dimension of evaluation beyond existing benchmarks, for their growing use in reliability-sensitive applications.

Run-level pass rate overstates retry-free coverage by up to 17.8 percentage points -- and the gap is largest precisely for mid-performing systems. We investigate this accuracy--stability relationship in large language model (LLM) evaluation for deterministic text-conditioned generation, using programming tasks as a concrete testbed. Standard code-generation benchmarks emphasize single-run accuracy or eventual success under repeated sampling, but many deployment settings also require stability: consistent outcomes across repeated invocations under the same task description. We present a repeated-run evaluation protocol with metrics for run-level accuracy, retry-free coverage, and per-problem variability. On a recency-based benchmark of 100 LeetCode-style problems, we evaluate 16 models from five provider families under two prompt templates with five repeated runs per problem, yielding 16,000 evaluation instances. Although run-level pass rate and perfect stability rate are strongly correlated (r=0.985), pass rate consistently exceeds retry-free coverage -- a gap that reaches 17.8 percentage points and reverses model rankings even among closely matched systems. Prompt effects are model-dependent rather than uniformly beneficial. These results suggest that repeated-run stability analysis is a necessary complement to conventional accuracy reporting for deterministic text-conditioned generation tasks.