Sharp Bounds on the Winner's Bias under Independent and Nonnegatively Correlated Noise

Reusing the scores that select a Best-of-$N$ winner can overstate its expected reward. We study evaluation from a fixed matrix of $K$ independent scores per candidate for a policy that selects using $J$ fresh scores. A single estimator based only on this matrix is exactly unbiased for expected judge reward under every independent, stable collection of candidate-specific score laws if and only if $J<K$, for every pool size $M\ge N\ge2$. At $J=K-1$, the selector deepens as $K$ grows. For independent Gaussian scores with common variance and fixed $M\ge N\ge2$, the unbiased minimax risk in this regime is of order $σ^2/\sqrt K$, attained by Holdout; allowing bias improves the rate to $σ^2/K$. For two candidates, we derive the minimum-variance unbiased estimator at known variance and the sharp asymptotic unbiased minimax constant $1/(π\sqrt2)$, which Holdout attains without knowing the variance. The cyclic average over subsets and ties can be computed in $O(MK\log M)$ operations. At fixed selector depth, cyclic evaluation of bounded scores has $O(K^{-1})$ risk uniformly in pool size. The impossibility result concerns the fixed matrix: one additional fresh winner score permits unbiased evaluation of the all-$K$ policy.

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.

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.