For centuries, it has been widely believed that the influence of a small coalition of voters is negligible in a large election. Consequently, there is a large body of literature on characterizing the likelihood for an election to be influenced when the votes follow certain distributions, especially the likelihood of being manipulable by a single voter under the i.i.d. uniform distribution, known as the Impartial Culture (IC). In this paper, we extend previous studies in three aspects: (1) we propose a more general semi-random model, where a distribution adversary chooses a worst-case distribution and then a contamination adversary modifies up to $\psi$ portion of the data, (2) we consider many coalitional influence problems, including coalitional manipulation, margin of victory, and various vote controls and bribery, and (3) we consider arbitrary and variable coalition size $B$. Our main theorem provides asymptotically tight bounds on the semi-random likelihood of the existence of a size-$B$ coalition that can successfully influence the election under a wide range of voting rules. Applications of the main theorem and its proof techniques resolve long-standing open questions about the likelihood of coalitional manipulability under IC, by showing that the likelihood is $\Theta\left(\min\left\{\frac{B}{\sqrt n}, 1\right\}\right)$ for many commonly-studied voting rules. The main technical contribution is a characterization of the semi-random likelihood for a Poisson multinomial variable (PMV) to be unstable, which we believe to be a general and useful technique with independent interest.

翻译：摘要：几个世纪以来，人们普遍认为，在大规模选举中，选民小联盟的影响力微乎其微。因此，大量文献致力于描述当选票遵循特定分布（特别是独立同分布均匀分布，即无偏文化（IC））时选举被影响的概率，尤其是单个选民在IC下操纵选举的可能性。本文从三个方面扩展了先前的研究：（1）我们提出了一种更一般的半随机模型，其中分布对手选择最坏情况分布，然后污染对手修改至多ψ比例的数据；（2）我们考虑多种联盟影响问题，包括联盟操纵、获胜边际以及各种选票控制和贿赂问题；（3）我们考虑任意且可变的联盟规模B。我们的主要定理在多种投票规则下，为存在规模为B且能成功影响选举的联盟的半随机概率提供了渐近紧界。该定理及其证明技术的应用解决了IC下联盟可操纵概率的长期未决问题，表明对于许多常见投票规则，该概率为Θ(min{B/√n, 1})。主要技术贡献在于刻画了泊松多项分布变量不稳定的半随机概率，我们认为这是一种具有独立价值的通用且有用的技术。