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Geometric Ways of Understanding Voting Problems

  • Author(s): McIntee, Tomas
  • Advisor(s): Saari, Donald
  • et al.
Abstract

General conclusions relating pairwise tallies with positional (e.g., plurality, 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 there must be a profile whereby each candidate is the ``winner'' with some positional rule. Previous work relating the probability of positional and pairwise tallies have used specific selected distributions (primarily the Impartial Culture and Impartial Anonymous Culture assumptions) and specific voting rules (particularly plurality). Techniques are developed here that can be applied to analyzing the probability of conflict between all different positional methods, and between combinations of pairwise tallies with positional results. Results are given for several broad categories of probability distribution, along with a qualitative analysis of the relationship between probability distributions over voter profiles and the likelihood of voting paradoxes. A method of geometrically comparing multiple-stage and single-stage elections is developed, which shows that multiple stage elections are not necessarily more vulnerable to being manipulated, but less vulnerable when all rank-order outcomes matter, and specifically only similar when an election only identifies a first-place winner. In the case where results are defined in terms of a singular winner, a plurality vote is identified as less manipulable in a single stage than in multiple stages, while an antiplurality vote is identified as more vulnerable in a single stage than in multiple stages.

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