Separating and Collapsing Electoral Control Types

Electoral control refers to attacking elections by adding, deleting, or partitioning voters or candidates [3]. Hemaspaandra et al. [16] discovered, for seven pairs (T , T ′) of seemingly distinct standard electoral control types, that T and T ′ are identical: For each input 𝐼 and each election system E, 𝐼 is a "yes" instance of both T and T ′ under E, or of neither. Surprisingly, this had gone undetected even as the field was score-carding how many standard 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 standard 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 seven. For three central election systems, we determine which control pairs are identical ("collapse") with respect to those particular systems, and we explore containment/incomparability relationships between control pairs. For approval voting, which has a different "type" for its votes, Hemaspaandra et al.'s [16] seven 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 one additional collapse for veto and none for plurality. We prove that each of the three election systems mentioned have no collapses other than those inherited from Hemaspaandra et al. [16] or added here. But we show many new containment relationships that hold between some separating control pairs, and for each separating pair of standard 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 three important election systems, clarifies the landscape of the 44 standard control types, for each pair collapsing or separating them, and also providing finer-grained information on the separations.