The \(q\)-majority efficiency of positional rules

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