Complexity and the geometry of voting

Geometry is used to analyze the surprisingly complex topic of “voting:” It is used to find classes of profiles where different rules have conflicting outcomes, to uncover new voting paradoxes, to determine the likelihood of disagreement among election outcomes, to explain classical mysteries such as the “paradox of voting,” and to analyze a variety of seemingly disparate topics such as strategic behavior, monotonicity, and the “no-show” paradox. Another geometric approach identifies all possible positional and Approval Voting election outcomes generated by a profile: the converse creates a geometric tool to identify new election relationships. A geometric “profile decomposition” identifies and explains all possible differences in positional and pairwise voting outcomes.