CONNECTING PAIRWISE AND POSITIONAL ELECTION OUTCOMES DONALD G. SAARI AND TOMAS J. MCINTEE INSTITUTE FOR MATHEMATICAL BEHAVIORAL SCIENCE UNIVERSITY OF CALIFORNIA, IRVINE, CA 92697-5100 Abstract. General conclusions relating pairwise tallies with positional (e.g., plural- ity, antiplurality (“vote-for-two”)) election outcomes were previously known only for the Borda Count. While it has been known since the eighteenth century that the Borda and Condorcet winners need not agree, it had not been known, for instance, in which settings the Condorcet and plurality winners can disagree, or must agree. Results of this type are developed here for all three-alternative positional rules. These relationships are based on an easily used method that connects pairwise tallies with admissible positional outcomes; e.g., a special case provides the first necessary and sufficient conditions ensuring that the Condorcet winner is the plurality winner; another case identifies when a profile must exist whereby each candidate is the “winner” with a specific positional rule. 1. Introduction After a quarter of a millennium of study, it is clear that the objective of determining which voting method most accurately reflects the views of the voters is a surprisingly subtle, major challenge. The complexity of this issue has forced researchers to adopt secondary measures, such as seeking properties of specific rules or probability estimates of paradoxical events. While providing useful information, these approaches remain surrogates for the true intent of identifying which profiles cause different kinds of election outcomes. Rather than determining the likelihood of particular paradoxical outcomes, for instance, a preferred outcome would be to identify all profiles that cause these difficulties. To advance our understanding of which profiles create various conclusions, the approach introduced here identifies all three-alternative profiles that support specified paired major- ity vote tallies. An advantage of knowing all possible supporting profiles is that it now is possible to determine all of the associated positional outcomes. To illustrate the variety of new questions that can be answered, suppose all we know about a profile is that its majority vote pairwise comparisons are A beats B by 70:30, A beats C by 60:40, and B beats C by 55:45. Here A is the Condorcet winner (she beats all other candidates) and C is the Condorcet loser (she loses to everyone). Just from these tallies, where the two involving the Condorcet winner A are of “landslide proportions” (winning 60% or more of the vote), the goal Our thanks to a referee who made several very useful suggestions that improved our presentation. Saari’s research was supported by NSF CMMI-1016785. 1