Voting procedures with incomplete preferences

We extend the application of a voting procedure (usually defined on complete preference relations over candidates) when the voters’ preferences consist of partial orders. We define possible (resp. necessary) winners for a given partial preference profile R with respect to a given voting procedure as the candidates being the winners in some (resp. all) of the complete extensions of R. We show that, although the computation of possible and necessary winners may be hard in general case, it is polynomial for the family of positional scoring procedures. We show that the possible and necessary Condorcet winners for a partial preference profile can be computed in polynomial time as well. Lastly, we point out connections to vote manipulation and elicitation.