Basic Geometry of Voting

I. From an Election Fable to Election Procedures.- 1.1 An Electoral Fable.- 1.1.1 Time for the Dean.- 1.1.2 The Departmental Election.- 1.1.3 Exercises.- 1.2 The Moral of the Tale.- 1.2.1 The Basic Goal.- 1.2.2 Other Political Issues.- 1.2.3 Strategic Behavior.- 1.2.4 Some Procedures Are Better than Others.- 1.2.5 Exercises.- 1.3 From Aristotle to “Fast Eddie”.- 1.3.1 Selecting a Pope.- 1.3.2 Procedure Versus Process.- 1.3.3 Jean-Charles Borda.- 1.3.4 Beyond Borda.- 1.4 What Kind of Geometry?.- 1.4.1 Convexity and Linear Mappings.- 1.4.2 Convex Hulls.- 1.4.3 Exercises.- II. Geometry for Positional And Pairwise Voting.- 2.1 Ranking Regions.- 2.1.1 Normalized Election Tally.- 2.1.2 Ranking Regions.- 2.1.3 Exercises.- 2.2 Profiles and Election Mappings.- 2.2.1 The Election Mapping.- 2.2.2 The Geometry of Election Outcomes.- 2.2.3 Exercises.- III. The Problem With Condorcet.- 3.1 Why Can’t an Organization Be More Like a Person?.- 3.1.1 Confused, Irrational Voters.- 3.1.2 Information Lost from Pairwise Majority Voting.- 3.1.3 Reduced Profiles.- 3.1.4 Exercises.- 3.2 Geometry of Pairwise Voting.- 3.2.1 The Geometry of Cycles.- 3.2.2 Cyclic Profile Coordinates.- 3.2.3 Power of Cyclic Coordinates.- 3.2.4 The Return of Confused Voters.- 3.2.5 Exercises.- 3.3 Black’s Single-Peakedness.- 3.3.1 Black’s Condition.- 3.3.2 Condorcet Winners and Losers.- 3.3.3 A Condorcet Improvement.- 3.3.4 Exercises.- 3.4 Arrow’s Theorem.- 3.4.1 A Sen Type Theorem.- 3.4.2 Universal Domain any IIA.- 3.4.3 Involvement and Voter Re ponsiveness.- 3.4.4 Arrow’s Theorem.- 3.4.5 A Dictatorship or an Informational Problem?.- 3.4.6 Elementary Algebra.- 3.4.7 The Fci,cj Level Sets.- 3.4.8 Some Existence Theorems.- 3.4.9 Intensity IIA.- 3.4.10 Exercises.- IV. Positional Voting And the BC.- 4.1 Positional Voting Methods.- 4.1.1 The Difference a Procedure Makes.- 4.1.2 An Equivalence Relationship for Voting Vectors.- 4.1.3 The Geometry of ws Outcomes.- 4.1.4 Exercises.- 4.2 What a Difference a Procedure Makes; Several Different Outcomes.- 4.2.1 How Bad It Can Get.- 4.2.2 Properties of Sup(p).- 4.2.3 The Procedure Line.- 4.2.4 Using the Procedure Line.- 4.2.5 Robustness of the Paradoxical Assertions.- 4.2.6 Proofs.- 4.2.7 Exercises.- 4.3 Positional Versus Pairwise Voting.- 4.3.1 Comparing Votes With a Fat Triangle.- 4.3.2 Positional Group Coordinates.- 4.3.3 Profile Sets.- 4.3.4 Some Comparisons.- 4.3.5 Comparisons.- 4.3.6 How Varied Does It Get?.- 4.3.7 Exercises.- 4.4 Profile Decomposition.- 4.4.1 Neutrality and Reversal Bias.- 4.4.2 Reversal Sets.- 4.4.3 Cancellation.- 4.4.4 Basic Profiles.- 4.4.5 Symmetry of Voting Vectors.- 4.4.6 Exercises.- 4.5 From Aggregating Pairwise Votes to the Borda Count.- 4.5.1 Borda and Aggregated Pairwise Votes.- 4.5.2 Basic Profiles.- 4.5.3 Geometric Representation.- 4.5.4 The Borda Dictionary.- 4.5.5 Borda Cross-Sections.- 4.5.6 Exercises.- 4.6 The Other Positional Voting Methods.- 4.6.1 What Can Accompany a F3 Tie Vote?.- 4.6.2 A Profile Coordinate Representation Approach.- 4.6.3 What Pairwise Outcomes Can Accompany a ws Tally?.- 4.6.4 Probability Computations.- 4.6.5 Exercises.- 4.7 Multiple Voting Schemes.- 4.7.1 From Multiple Methods to Approval Voting.- 4.7.2 No Good Deed Goes Unpunished.- 4.7.3 Comparisons.- 4.7.4 Averaged Multiple Voting Systems.- 4.7.5 Procedure Strips.- 4.7.6 Exercises.- 4.8 Other Election Procedures.- 4.8.1 Other Pairwise Procedures.- 4.8.2 Runoffs.- 4.8.3 Scoring Runoffs.- 4.8.4 Comparisons of Positional Voting Outcomes.- 4.8.5 Plurality or a Runoff?.- 4.8.6 Exercises.- V. Other Voting Issues.- 5.1 Weak Consistency: The Sum of the Parts.- 5.1.1 Other Uses of Convexity.- 5.1.2 An L of an Agenda.- 5.1.3 Condorcet Extensions.- 5.1.4 Other Pairwise Procedures.- 5.1.5 Maybe “If’s “ and “And’s”, But No “Or’s” or “But’s”.- 5.1.6 A General Theorem.- 5.1.7 Exercises.- 5.2 From Involvement and Monotonicity to Manipulation.- 5.2.1 Positively Involved.- 5.2.2 Monotonicity.- 5.2.3 A Profile Angle.- 5.2.4 A General Theorem Using Profiles.- 5.2.5 Other Admissible Directions.- 5.2.6 Exercises.- 5.3 Gibbard-Satterthwaite and Manipulable Procedures.- 5.3.1 Measuring Suspectibility to Manipulation.- 5.3.2 Exercises.- 5.4 Proportional Representation.- 5.4.1 Hare and Single Transferable Vote.- 5.4.2 The Apportionment Problem.- 5.4.3 Something Must Go Wrong — Alabama Paradox.- 5.4.4 A Better Improved Method?.- 5.4.5 More Surprises, But Not Problems.- 5.4.6 Exercises.- 5.5 House Monotone Methods.- 5.5.1 Who Cares About Quota?.- 5.5.2 Big States, Small States.- 5.5.3 The Translation Bias.- 5.5.4 Sliding Bias.- 5.5.5 If Washington Had More People 279.- 5.5.6 A Solution.- 5.5.7 Exercises.- VI. Notes.- VII. References.