Abstract
Many studies have evaluated the probabilities that weighted scoring rules (WSRs) select the Condorcet winner or match the complete ranking by pairwise majority rule (PMR) in three candidate elections. This study considers situations in which PMR cycles exist, to evaluate the associated probabilities that WSRs select the Condorcet winner or match the complete ranking of a modified PMR relation. This modified relation reverses the PMR relationship on the pair with the smallest majority rule advantage. Evidence indicates that the WSRs that maximize these probabilities when a Condorcet winner exists also maximize the associated probabilities when a PMR cycle exists.
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