The $$q$$q-majority efficiency of positional rules

According to a given quota $$q$$q, a candidate $$a$$a is beaten by another candidate $$b$$b if at least a proportion of $$q$$q individuals prefer $$b$$b to $$a$$a. The $$q$$q-majority efficiency of a voting rule is the probability that the rule selects a candidate who is never beaten under the $$q$$q-majority, given that such a candidate exits. Closed form representations are obtained for the $$q$$q-majority efficiency of positional rules (simple and sequential) in three-candidate elections. It turns out that the $$q$$q-majority efficiency is: (i) significantly greater for sequential rules than for simple positional rules; and (ii) very close to the $$q$$q-Condorcet efficiency, the conditional probability that a positional rule will elect the candidate who beats all others under the $$q$$q-majority, when one exists.