Neutral candidates in approval and disapproval vote

In this article, the question is to select the “best” candidates within a set of candidates when voters cast approval-disapproval ternary ballots. That is, three options are offered to voters: casting a vote “in favor”, a “neutral” vote or a vote “against” each candidate. We first review desirable properties that a rule aggregating approval-disapproval ternary ballots should satisfy. We check whether the rules that have been proposed in the literature satisfy them. Then, we provide comparable axiomatizations of three rules: one is the lexicographical extension of the Approval rule for binary ballots; the second is the lexicographical extension of the Disapproval rule for binary ballots; and the third rule eliminates candidates with more opponents and fewer supporters than other candidates.