Copeland Voting Fully Resists Constructive Control

Abstract

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Faliszewski et al. [9] 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 Copeland^α, 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 Copeland^α 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 Copeland^α is computationally resistant to control in that scenario (i.e., we prove that control problem to be NP-hard). In particular, we prove that Copeland^0.5, 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 Copeland^α (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”).

Supported in part by DFG grants RO-1202/9-3 and RO-1202/11-1, NSF grants CCR-0311021, CCF-0426761, and IIS-0713061, the Alexander von Humboldt Foundation’s TransCoop program, and two Friedrich Wilhelm Bessel Research Awards. URLs: www.cs.rochester.edu/u/{pfali, lane} (Piotr Faliszewski and Lane A. Hemaspaandra), www.cs.rit.edu/~ eh (Edith Hemaspaandra), and ccc.cs.uni-duesseldorf.de/~ rothe (Jörg Rothe). Corresponding author: Jörg Rothe.