Chapter Twenty-Seven - Geometry of Voting

It is shown how simple geometry can be used to analyze and discover new properties about pairwise and positional voting rules as well as for those rules (e.g., runoffs and Approval Voting) that rely on these methods. The description starts by providing a geometric way to depict profiles, which simplifies the computation of the election outcomes. This geometry is then used to motivate the development of a “profile coordinate system,” which evolves into a tool to analyze voting rules. This tool, for instance, completely explains various longstanding “paradoxes,” such as why a Condorcet winner need not be elected with certain voting rules. A different geometry is developed to indicate whether certain voting “oddities” can be dismissed or must be taken seriously, and to explain why other mysteries, such as strategic voting and the no-show paradox (where a voter is rewarded by not voting), arise. Still another use of geometry extends McGarvey's Theorem about possible pairwise election rankings to identify the actual tallies that can arise (a result that is needed to analyze supermajority voting). Geometry is also developed to identify all possible positional and Approval Voting election outcomes that are admitted by a given profile; the converse becomes a geometric tool that can be used to discover new election relationships. Finally, it is shown how lessons learned in social choice, such as the seminal Arrow's and Sen's Theorems and the expanding literature about the properties of positional rules, provide insights into difficulties that are experienced by other disciplines.