When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation
Abstract
Increasing a voting budget can improve or reduce population accuracy, and an initial gain or loss need not persist. We give a complete structural classification of these curves for binary unweighted majority under infinitely exchangeable correctness indicators, with an arbitrary latent success-probability law Π. The classification is encoded by the signed voting signature $ ω_Π(B)=\int \mathbf{1}\{q(1-q)\in B\}(2q-1)\,Π(\,\mathrm{d} q). $ The complete odd-budget curve and this signature determine each other: the initial accuracy gives its total signed mass, and successive budget increments give its higher moments. Reflection-symmetric differences between latent laws are exactly the changes invisible to voting. A weighted total-variation constraint characterizes every attainable signature, and a minimum-mass allocation to the two reflected branches, completed by symmetric probability mass, gives every population producing its curve. Consequently, the full curve identifies the latent population precisely when no symmetric mass remains to redistribute.
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BibTeX
@misc{liu2026when,
title = {{When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation}},
author = {Yi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.05592},
eprint = {2605.05592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05592},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - When Can Voting Help, Hurt, or Change Course? Exact Structure of Binary Test-Time Aggregation
AU - Yi Liu
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2605.05592
UR - https://arxiv.org/abs/2605.05592
AB - Increasing a voting budget can improve or reduce population accuracy, and an initial gain or loss need not persist. We give a complete structural classification of these curves for binary unweighted majority under infinitely exchangeable correctness indicators, with an arbitrary latent success-probability law Π. The classification is encoded by the signed voting signature $ ω_Π(B)=\int \mathbf{1}\{q(1-q)\in B\}(2q-1)\,Π(\,\mathrm{d} q). $ The complete odd-budget curve and this signature determine each other: the initial accuracy gives its total signed mass, and successive budget increments give its higher moments. Reflection-symmetric differences between latent laws are exactly the changes invisible to voting. A weighted total-variation constraint characterizes every attainable signature, and a minimum-mass allocation to the two reflected branches, completed by symmetric probability mass, gives every population producing its curve. Consequently, the full curve identifies the latent population precisely when no symmetric mass remains to redistribute.
ER -