[HHM20] discovered, for 7 pairs (C,D) of seemingly distinct standard electoral control types, that C and D are identical: For each input I and each election system, I is a Yes instance of both C and D, or of neither. Surprisingly this had gone undetected, even as the field was score-carding how many std. control types election systems were resistant to; various "different" cells on such score cards were, unknowingly, duplicate effort on the same issue. This naturally raises the worry that other pairs of control types are also identical, and so work still is being needlessly duplicated. We determine, for all std. control types, which pairs are, for elections whose votes are linear orderings of the candidates, always identical. We show that no identical control pairs exist beyond the known 7. We for 3 central election systems determine which control pairs are identical ("collapse") with respect to those systems, and we explore containment/incomparability relationships between control pairs. For approval voting, which has a different "type" for its votes, [HHM20]'s 7 collapses still hold. But we find 14 additional collapses that hold for approval voting but not for some election systems whose votes are linear orderings. We find 1 additional collapse for veto and none for plurality. We prove that each of the 3 election systems mentioned have no collapses other than those inherited from [HHM20] or added here. But we show many new containment relationships that hold between some separating control pairs, and for each separating pair of std. control types classify its separation in terms of containment (always, and strict on some inputs) or incomparability. Our work, for the general case and these 3 important election systems, clarifies the landscape of the 44 std. control types, for each pair collapsing or separating them, and also providing finer-grained information on the separations.

翻译：[HHM20] 发现，对于7对看似不同的标准选举控制类型(C,D)，C和D实际上是相同的：对于任意输入I和任意选举系统，I要么同时是C和D的肯定实例，要么同时不是。令人惊讶的是，这一事实此前未被察觉，尽管该领域一直在统计选举系统能抵抗多少种标准控制类型；此类统计表中各种“不同”的单元格实际上是在同一问题上无谓的重复劳动。这自然引发了担忧：其他控制类型对可能也是相同的，因此研究工作仍在被不必要地重复。我们针对所有标准控制类型，在选票为候选人线性排序的选举中，确定了哪些类型对总是相同的。我们证明，除了已知的7对外，不存在其他相同的控制类型对。我们针对3个核心选举系统，确定了哪些控制类型对在这些系统下是相同的（即“坍缩”），并探讨了控制类型对之间的包含/不可比关系。对于具有不同选票“类型”的赞同投票，[HHM20]发现的7个坍缩仍然成立。但我们发现了14个额外的坍缩，这些坍缩适用于赞同投票，但不适用于某些选票为线性排序的选举系统。我们为否决投票发现了1个额外坍缩，为简单多数制则未发现新增坍缩。我们证明，上述3种选举系统除了继承自[HHM20]或本文新增的坍缩外，不存在其他坍缩。但我们展示了许多存在于某些分离控制类型对之间的新包含关系，并对每个分离的标准控制类型对，根据包含关系（总是成立，且在部分输入上严格成立）或不可比性对其分离程度进行了分类。我们的工作，针对一般情况以及这3个重要的选举系统，厘清了44种标准控制类型的格局，明确了每对类型是坍缩还是分离，并为分离情况提供了更细粒度的信息。