Basic Geometry of Voting

Amazingly, the complexities of voting theory can be explained and resolved with comfortable geometry. A geometry which unifies such seemingly disparate topics as manipulation, monotonicity, and even the apportionment issues of the US Supreme Court. Although directed mainly toward students and others wishing to learn about voting, experts will discover here many previously unpublished results. As an example, a new profile decomposition quickly resolves the age-old controversies of Condorcet and Borda, demonstrates that the rankings of pairwise and other methods differ because they rely on different information, casts serious 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.

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.