Llull and Copeland Voting Computationally Resist Bribery and Control
Abstract
The only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. We study a parameterized version of Copeland voting, denoted by Copelandα, where the parameter α is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. We prove that Copeland0.5, the system commonly referred to as "Copeland voting," provides full resistance to constructive control, and we prove the same for Copelandα, for all rational α, 0 < α < 1. Copeland voting is the first natural election system proven to have full resistance to constructive control. We also prove that both Copeland1 (Llull elections) and Copeland0 are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational α, 0 ≤ α ≤ 1, Copelandα voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copelandα. We also study Copelandα elections under more flexible models such as microbribery and extended control and we integrate the potential irrationality of voter preferences into many of our results.