Geometry of Voting

Over two centuries of theory and practical experience have taught us that election and decision procedures do not behave as expected. Instead, we now know that when different tallying methods are applied to the same ballots, radically different outcomes can emerge, that most procedures can select the candidate, the voters view as being inferior, and that some commonly used methods have the disturbing anomaly that a winning candidate can lose after receiving added support. A geometric theory is developed to remove much of the mystery of three-candidate voting procedures. In this manner, the spectrum of election outcomes from all positional methods can be compared, new flaws with widely accepted concepts (such as the "Condorcet winner") are identified, and extensions to standard results (e.g. Black's single-peakedness) are obtained. Many of these results are based on the "profile coordinates" introduced here, which makes it possible to "see" the set of all possible voters' preferences leading to specified election outcomes. Thus, it now is possible to visually compare the likelihood of various conclusions. Also, geometry is applied to apportionment methods to uncover new explanations why such methods can create troubling problems.

Inhalt
I. From an Election Fable to Election Procedures.- 1.1 An Electoral Fable.- 1.2 The Moral of the Tale.- 1.3 From Aristotle to Fast Eddie.- 1.4 What Kind of Geometry.- II. Geometry for Positional and Pairwise Voting.- 2.1 Ranking Regions.- 2.2 Profiles and Election Mappings.- 2.3 Positional Voting Methods.- 2.4 What a Difference a Procedure Makes; Several Different Outcomes.- 2.5 Why Can't an Organization Be More Like a Person?.- 2.6 Positional Versus Pairwise Voting.- III. From Symmetry to the Borda Count and Other Procedures.- 3.1 Symmetry.- 3.2 From Aggregating Pairwise Votes to the Borda Count.- 3.3 The Other Positional Voting Methods.- 3.4 Multiple Voting Schemes.- 3.5 Other Election Procedures.- IV. Many Profiles; Many New Paradoxes.- 4.1 Weak Consistency: The Sum of the Parts.- 4.2 From Involvement and Monotonicity to Manipulation.- 4.3 Proportional Representation.- 4.4 Arrow's Theorem.- 4.5 Characterizations of Scoring, Positional and Borda.- Notes.- References.