How Many Humans Are 32 LLM Judges Worth?

Abstract

A panel's human-equivalent size is target-specific. Matching a fixed 32-judge panel to empirical human label distributions on three ChaosNLI tasks yields two distinct effective sizes: distributional-error matching gives $\nu_{\mathrm{MSE}}=2.304$, $3.750$, and $3.445$, whereas spectral matching gives $\nu_H=4.242$, $6.459$, and $6.499$, a gap of $1.72$--$1.89\times$; a binary-error diagnostic credits the same panels with only $1.971$--$2.227$ effective votes. Extrapolating the distributional-error curve at fixed squared mean residual, mean member variance, and normalized mean covariance gives asymptotes of $2.392$, $3.990$, and $3.655$, with 32 judges already reaching $94.0$--$96.3\%$. An exact spectral identity explains the gap: error depends on member energy and on the orientation of residual variation relative to averaging, information that the participation ratio (PR) discards. A realizable hard-label construction confirms that higher spectral diversity can coexist with worse distribution recovery even under equal member energies and nonnegative correlations, and the consensus direction retains $\gamma_{\mathrm{co}}=43.8\%$, $33.7\%$, and $35.9\%$ of centered residual variance. An external check on CC-1000, a 1,000-item Civil Comments subset with a different panel, gives $\nu_H=2.84$. For panel choice, we establish an existence result and one feasible path: exhaustive enumeration at $k\in\{5,7\}$ shows that panels beating the accuracy-top-$k$ baseline on both accuracy and $\nu_H$ always exist, and greedily swapping at most two members reaches $24.8$--$56.0\%$ higher $\nu_H$ at $0.10$--$1.10$ percentage points higher accuracy. Our dataset and code are available at https://github.com/Chao1208/32judges-votes.