Complexity theory is a useful tool to study computational issues surrounding the elicitation of preferences, as well as the strategic manipulation of elections aggregating together preferences of multiple agents. We study here the complexity of determining when we can terminate eliciting preferences, and prove that the complexity depends on the elicitation strategy. We show, for instance, that it may be better from a computational perspective to elicit all preferences from one agent at a time than to elicit individual preferences from multiple agents. We also study the connection between the strategic manipulation of an election and preference elicitation. We show that what we can manipulate affects the computational complexity of manipulation. In particular, we prove that there are voting rules which are easy to manipulate if we can change all of an agent's vote, but computationally intractable if we can change only some of their preferences. This suggests that, as with preference elicitation, a fine-grained view of manipulation may be informative. Finally, we study the connection between predicting the winner of an election and preference elicitation. Based on this connection, we identify a voting rule where it is computationally difficult to decide the probability of a candidate winning given a probability distribution over the votes.
Control of elections refers to attempts by an agent to, via such actions as addition deletion partition of candidates or voters, ensure that a given candidate wins (Bartholdi, Tovey, & Trick 1992). An election system in which such an agent's computational task is NP-hard is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems known to be resistant to all the standard control types are highly artificial election systems created by hybridization (Hemaspaandra, Hemaspaandra, & Rothe 2007b). In this paper, we prove that an election system developed by the 13th century mystic Ramon Llull and the well-studied Copeland election system are both resistant to all the standard types of (constructive) electoral control other than one variant of addition of candidates. This is the most comprehensive resistance to control yet achieved by any natural election system whose winner problem is in P. In addition, we show that Llull and Copeland voting are very broadly resistant to bribery attacks, and we integrate the potential irrationality of voter preferences into many of our results.