[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，在7对看似不同的标准选举控制类型(C,D)中，C和D实际上是相同的：对于每个输入I和每个选举系统，I要么同时是C和D的Yes实例，要么两者都不是。令人惊讶的是，这一现象此前未被察觉，即使该领域一直在为选举系统能抵抗多少种标准控制类型进行评分；这些评分表上各种"不同"的单元格实际上在无意识中重复研究了同一问题。这自然引发担忧：其他控制类型对也可能相同，因此仍然存在不必要的重复工作。我们针对所有标准控制类型，确定了在选票为候选人线性排序的选举中，哪些对总是相同的。我们证明，除了已知的7对之外，不存在其他相同的控制类型对。针对3个核心选举系统，我们确定了哪些控制类型对在这些系统中是相同的（即"合并"），并探索了控制类型对之间的包含/不可比较关系。对于采用不同选票"类型"的同意投票制，[HHM20]的7个合并仍然成立。但我们发现同意投票制额外存在14个合并，而这些合并对于那些选票为线性排序的选举系统并不成立。我们在否决投票制中发现1个额外合并，而在简单多数制中未发现额外合并。我们证明，上述3个选举系统除了继承自[HHM20]或本文新增的合并外，不存在其他合并。但我们证明了许多新的包含关系，这些关系存在于某些可分离的控制类型对之间，并且对于每对可分离的标准控制类型，我们根据包含关系（总是成立，且在某些输入上严格成立）或不可比较关系对其分离性进行分类。我们的工作针对一般情况以及这3个重要选举系统，阐明了44种标准控制类型的整体格局，为每对控制类型提供了合并或分离的判定，并提供了关于分离性的更精细信息。