Large player classes and approximately weighted voting
Abstract
It is known that completeness of the coalitional (C-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an n-player simple game ε-weighted if, for some weight vector in the (n−1)-simplex, any losing coalition weight exceeds any winning coalition weight by at most ε; let ε∗ be the smallest such ε. We establish an upper bound on ε∗ in a general C-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then ε∗=O(1/nmin) for the minimal player class size nmin, and the bound is tight. Importantly, the minimal attainable fraction δ of coalitions mislabeled by a weighting rule also tends to $0$ when nmin→∞ and the number of player classes is bounded, with δ=O(1/√(nmin)).
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BibTeX
@misc{einy2026large,
title = {{Large player classes and approximately weighted voting}},
author = {Ezra Einy and Ori Haimanko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08968},
eprint = {2610.08968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08968},
}
RIS
TY - UNPB
TI - Large player classes and approximately weighted voting
AU - Ezra Einy
AU - Ori Haimanko
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.08968
UR - https://arxiv.org/abs/2610.08968
AB - It is known that completeness of the coalitional (C-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an n-player simple game ε-weighted if, for some weight vector in the (n−1)-simplex, any losing coalition weight exceeds any winning coalition weight by at most ε; let ε^∗ be the smallest such ε. We establish an upper bound on ε^∗ in a general C-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then ε^∗=O(1/n_min) for the minimal player class size n_min, and the bound is tight. Importantly, the minimal attainable fraction δ of coalitions mislabeled by a weighting rule also tends to $0$ when n_min→∞ and the number of player classes is bounded, with δ=O(1/√(n_min)).
ER -