Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Faliszewski et al.  proved that Llull voting (which is here denoted by Copeland1) and a variant (here denoted by Copeland0) of Copeland voting are computationally resistant to many, yet not all, types of constructive control and that they also provide broad resistance to bribery. 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 in Copeland elections. For each rational α, 0 < α< 1, and each previously studied control scenario, we either prove that Copelandα is computationally vulnerable to control in that scenario (i.e., we give a P-time algorithm that determines whether control is possible, and if so, determines exactly how to exert the control) or we prove that Copelandα is computationally resistant to control in that scenario (i.e., we prove that control problem to be NP-hard). In particular, we prove that Copeland0.5, the system commonly referred to as “Copeland voting,” provides full resistance to constructive control. Among systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. Looking at rational α, 0 < α< 1, we give a broad set of results on bribery and on the fixed-parameter tractability of bounded-case control for Copelandα (previously only Copeland0 and Copeland1 had been studied), and we introduce and obtain fixed-parameter tractability results even in a new, more flexible model of control (that we dub “extended control”).
Computational social choice theory preference aggregation multiagent systems