How Many Advisors Are Enough? Predicting Useful Majority-Vote Ensemble Size under Correlated Decision Errors

Abstract

Combining multiple imperfect advisors via majority vote is a standard way to improve decision reliability, but real advisors’ errors are correlated, which caps achievable accuracy and causes the benefit of additional advisors to saturate. Existing theory, effective-sample-size (ESS) approximations and correlated Condorcet jury results, establishes that this saturation occurs, but does not give a designer an inexpensive way to predict, from observable advisor accuracy and pairwise error correlation, how many advisors are actually worth adding. We address this gap with, a closed-form diagnostic derived from an exact Beta-Binomial model of exchangeable, correlated binary advisors, that identifies the ensemble size beyond which further majority-vote gains fall below a specified tolerance. We prove that the ESS baseline is exact in its variance term and that its divergence vanishes at chance-level accuracy, and we show this divergence is governed by an accuracy-correlation interaction driven by asymmetry rather than correlation alone. Against a 20-cell synthetic simulation grid, the proposed diagnostic tracks the observed useful ensemble size more closely than the ESS baseline, though the empirical advantage is only marginally significant given the Monte Carlo resolution achieved. The contribution is this predictive diagnostic and its theoretical relation to ESS (not the underlying saturation phenomenon, which is already known), and the scope is limited to exchangeable, homogeneous advisors under simple majority voting.