Determining winners in weighted sequential majority voting: incomplete profiles vs. majority graphs

In sequential majority voting, preferences are aggregated by a sequence of pairwise comparisons (also called an agenda) between candidates. The result of each comparison is determined by a weighted majority vote between the agents. In this paper we consider the situation where the agents may not have revealed all their preferences. This is common in real-life settings, due to privacy issues or an ongoing elicitation process. We study the computational complexity of determining the winner(s), given that some preferences may not yet be revealed and the agenda is not yet known or decided. We show that it is easy to determine if a candidate must win whatever the agenda. On the other hand, it is hard to know whether a candidate can win in at least one agenda for at least one completion of unrevealed preferences. This is also true if the agenda is balanced (that is, each candidate must win the same number of pairwise competitions to win overall). We also consider the case of fixed agendas. We show that in this case it is easy to determine if a candidate can win in the fixed agenda for at least a completion or for every completion of unrevealed preferences.