Basic Geometry of Voting

A surprise is how the complexities of voting theory can be explained and resolved with the comfortable geometry of our three-dimensional world. This book is directed toward students and others wishing to learn about voting, experts will discover previously unpublished results. As an example, a new profile decomposition quickly resolves two centuries old controversies of Condorcet and Borda, demonstrates, that the rankings of pairwise and other methods differ because they rely on different information, casts series doubt on the reliability of a Condorcet winner as a standard for the field, makes the famous Arrow`s Theorem predictable, and simplifies the construction of examples. The geometry unifies seemingly disparate topics as manipulation, monotonicity, and even the apportionment issues of the US Supreme Court.

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.- III. The Problem With Condorcet.- 3.1 Why Can’t an Organization Be More Like a Person?.- 3.2 Geometry of Pairwise Voting.- 3.3 Black’s Single-Peakedness.- 3.4 Arrow’s Theorem.- IV. Positional Voting And the BC.- 4.1 Positional Voting Methods.- 4.2 What a Difference a Procedure Makes; Several Different Outcomes.- 4.3 Positional Versus Pairwise Voting.- 4.4 Profile Decomposition.- 4.5 From Aggregating Pairwise Votes to the Borda Count.- 4.6 The Other Positional Voting Methods.- 4.7 Multiple Voting Schemes.- 4.8 Other Election Procedures.- V. Other Voting Issues.- 5.1 Weak Consistency: The Sum of the Parts.- 5.2 From Involvement and Monotonicity to Manipulation.- 5.3 Gibbard-Satterthwaite and Manipulable Procedures.- 5.4 Proportional Representation.- 5.5 House Monotone Methods.- VI. Notes.- VII. References.