Determining Possible and Necessary Winners Given Partial Orders

Abstract

Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of partial orders and a candidate c, two important questions arise: first, is c guaranteed to win, and second, is it still possible for c to win? These are the necessary winner and possible winner problems, respectively. We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We prove that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NP-complete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NP-complete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNP-complete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomial-time algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.