Control and manipulation are two of the most studied types of attacks on elections. In this paper, we study the complexity of control attacks on elections in which there are manipulators. We study both the case where the “chair” who is seeking to control the election is allied with the manipulators, and the case where the manipulators seek to thwart the chair. In the latter case, we see that the order of play substantially influences the complexity. We prove upper bounds, holding over every election system with a polynomial-time winner problem, for all standard control cases, and some of these bounds are at the second or third level of the polynomial hierarchy, and we provide matching lower bounds to prove these tight. Nonetheless, for important natural systems the complexity can be much lower. We prove that for approval and plurality elections, the complexity of even competitive clashes between a controller and manipulators falls far below those high bounds, even as low as polynomial time. Yet for a Borda-voting case we show that such clashes raise the complexity unless NP = coNP.

中文翻译：

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机械手在场时的控制：合作和竞争案例

控制和操纵是对选举的攻击中研究最多的两种类型。在本文中，我们研究了对具有操纵器的选举进行控制攻击的复杂性。我们研究了试图控制选举的“主席”与操纵者结盟的情况，以及操纵者试图挫败主席的情况。在后一种情况下，我们看到播放顺序会严重影响复杂性。对于所有标准控制案例，我们证明了上限，并用多项式时间获胜者问题保留了每个选举系统，并且其中一些界限位于多项式层次结构的第二或第三级，并且我们提供了匹配的下限来证明这些紧。但是，对于重要的自然系统而言，复杂度可能要低得多。我们证明，要获得批准和进行多次选举，控制器和操纵器之间甚至竞争性冲突的复杂性都远远低于那些上限，甚至低至多项式时间。但是，对于Borda投票案例，我们表明，除非NP = coNP，否则此类冲突会增加复杂性。