[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：对于每个输入I和每个选举系统，I要么同时是C和D的“是”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对抗多少种标准控制类型进行记分；这些记分卡上的各种“不同”单元格，实际上是在不知情的情况下对同一问题进行了重复工作。这自然引发担忧：其他控制类型对是否也相同，从而导致工作仍在被不必要地重复。我们针对所有标准控制类型，确定了在投票为候选人线性排序的选举中，哪些控制类型对总是相同的。我们证明，除已知的7对之外，不存在其他相同的控制类型对。我们针对三个核心选举系统，确定了哪些控制类型对相对于这些系统是相同的（即“合并”），并探讨了控制类型对之间的包含/不可比关系。对于具有不同投票“类型”的批准投票，[HHM20]中的7个合并仍然成立。但我们发现，批准投票还有14个额外的合并，这些合并不适用于某些投票为线性排序的选举系统。我们发现否决投票有1个额外合并，而多数投票则没有。我们证明，所提到的三个选举系统除了继承自[HHM20]或本文添加的合并外，没有其他合并。但我们展示了许多新的包含关系，这些关系存在于某些分离的控制类型对之间，并且对于每对分离的标准控制类型，我们根据包含关系（总是包含，或对某些输入严格包含）或不可比性对其分离进行分类。我们的工作，针对一般情况和这三个重要的选举系统，阐明了44种标准控制类型的格局，对每个控制类型对进行了合并或分离，并提供了关于分离的更细致信息。