Natural-language tasks can elicit different verdicts from protocol-following evaluators that receive the same declared information. We study aggregate disambiguation systems (ADSs). Given a task and a candidate solution, each evaluator casts a binary vote on whether the solution should be accepted, and the system aggregates the votes of a finite panel. The target is protocol reproducibility relative to an explicitly declared evaluator reference, not semantic truth. We separate fixed finite censuses, probabilistic evaluator populations, and growing-census limits, since their endpoint laws and guarantees are not interchangeable. In the population setting, we use finite samples to estimate how often a finite panel reaches the same decision as the declared evaluator population. We provide a lower confidence bound on the fraction of candidate solutions for which the disagreement probability is at most a chosen tolerance. The calculation accounts separately for sampling candidate solutions and sampling evaluators. The construction permits arbitrary dependence among columns induced by shared evaluator rows and uses exact binomial intervals at the evaluator layer and an exact one-sided binomial inversion at the generator layer. Simulations check the implementation against known population coverages and expose power limitations.
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.
Multiagent language model systems, such as collaborative reasoning and debate, produce multiple correlated candidate answers and confidence signals. However, these signals are usually calibrated only at the individual agent level, and provide no principled confidence estimate for the system's final answer. We formulate this as a confidence composition problem where combining confidence across agents and reasoning stages while preserving both selective utility and probabilistic reliability. We study confidence-aware routing and log-odds pooling protocols that select among candidate answers and output a system-level confidence. Across five benchmarks, 30 heterogeneous and homogeneous model pairs, and two confidence estimators, our gated-fusion methods improve AUARC and reduce Brier score over single agent, standard debate, and selective debate baselines, while retaining competitive weighted F1-score as a correctness metric. We further show that our log-odds fusion is overconfident due to correlated intermediate signals. We propose a shared dependence discount that substantially improves reliability while preserving predictions.
Language-model judges and reward models enable scalable supervision, but finite optimization can exploit evaluator errors rather than improve response quality. We characterize this failure through the covariance geometry of evaluator ensembles. For calibrated judges, the ensemble mean retains common-mode error along the all-ones direction, whereas cross-judge disagreement captures only orthogonal error. Consequently, disagreement can be high despite robust aggregation, or low while shared response-dependent errors persist. We prove that common-mode error is not identifiable from internal judge scores alone. Under a joint sub-Gaussian model, we bound best-of-K selection overstatement and target-quality regret, extending the guarantees to predictably adaptive search under conditional calibration. The resulting search terms scale as the square root of log K and are asymptotically tight for Gaussian projected errors. We further show that noisy quality proxies introduce artificial rank-one covariance without changing disagreement, and propose a bounded two-anchor Bernstein certificate for finite-search error and regret. Fixed-seed Gaussian stress tests over 120 (J, rho, K) configurations and real-model audits validate the theory while revealing the limits of disagreement-based diagnostics under increasing search pressure.