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：几个世纪以来，人们普遍认为在大型选举中，少数选民联盟的影响力微乎其微。因此，大量文献致力于刻画在选票服从特定分布时选举被影响的可能性，尤其是在独立同分布均匀分布（即无偏文化（Impartial Culture, IC））下，单个选民操纵选举的可能性。本文从三个方面拓展了先前研究：（1）提出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，随后污染对手修改最多$\psi$比例的数据；（2）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；（3）考虑任意且可变的联盟规模$B$。我们的主定理给出了在广泛投票规则下，存在大小为$B$的联盟能成功影响选举的半随机可能性的渐近紧界。该定理及其证明技巧的应用解决了关于IC下联盟操纵可能性的长期悬而未决的问题，表明对于许多常见投票规则，该可能性为$\Theta\left(\min\left\{\frac{B}{\sqrt n}, 1\right\}\right)$。主要技术贡献在于刻画了泊松多项变量（Poisson Multinomial Variable, PMV）不稳定的半随机可能性，我们相信这一通用且有用的技术具有独立的研究价值。