Strong measures of group coherence and the probability that a pairwise majority rule winner exists

Abstract

A Pairwise Majority Rule Winner (PMRW) exists for a voting situation if some candidate can defeat each of the remaining candidates by Pairwise Majority Rule. The PMRW would be very appropriate for selection as the winner of an election, but it is well known that such a candidate does not always exist. This paper concludes a series of studies regarding the probability that a PMRW should be expected to exist in three-candidate elections, by introducing the notion of a strong measures of mutually coherent group preferences. In order for voting situations to be reasonably expected to fail to have a PMRW in a three-candidate election, voters’ preferences must be generated in an environment that is far removed from the situation in which there is a strong-overall-unifying candidate. So far removed, that it is extremely unlikely that a PMRW will not exist in voting situations with large electorates for a small number of candidates.