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 tarrying 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 in this monograph 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 (Black's single-peakedness) are obtained. Many of these results are based on the profile co-ordinates introduced here, which make it possible to see the set of all possible voters' preferences leading to specified election outcomes. Geometry is applied to apportionment methods to uncover new explanations why such methods can create troubling problems.

Table of Contents:
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.- 1.2 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.- 1.3 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.- 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.- 2.3 Positional Voting Methods.- 2.3.1 The Difference a Procedure Makes.- 2.3.2 An Equivalence Relationship for Voting Vectors.- 2.3.3 The Geometry of w s Outcomes.- 2.3.4 Exercises.- 2.4 What a Difference a Procedure Makes; Several Different Outcomes.- 2.4.1 How Bad It Can Get.- 2.4.2 Properties of Sup(p).- 2.4.3 The Procedure Line.- 2.4.4 Using the Procedure Line.- 2.4.5 From Procedure Lines to Scoring Shells.- 2.4.6 Scoring Shell Geometry.- 2.4.7 Robustness of the Paradoxical Assertions.- 2.4.8 Proofs.- 2.4.9 Exercises.- 2.5 Why Can't an Organization Be More Like a Person?.- 2.5.1 Pairs and the Irrational Behavior of Organizations.- 2.5.2 Confused, Irrational Voters.- 2.5.3 Information Lost from Pairwise Majority Voting.- 2.5.4 Geometry of Pairwise Voting.- 2.5.5 The Geometry of Cycles.- 2.5.6 From Group Coordinates to Profile Restrictions.- 2.5.7 Black's Conditions for Avoiding Cycles.- 2.5.8 Spatial Voting.- 2.5.9 Extensions of Black's Condition.- 2.5.10 Condorcet Winners and Losers.- 2.5.11 A Condorcet Improvement.- 2.5.12 Exercises.- 2.6 Positional Versus Pairwise Voting.- 2.6.1 Comparing Votes with a Fat Triangle.- 2.6.2 Positional Group Coordinates.- 2.6.3 Profile Sets.- 2.6.4 Some Comparisons.- 2.6.5 How Likely Is It?.- 2.6.6 How Varied Does It Get?.- 2.6.7 Procedures Lines and Cyclic Coordinates.- 2.6.8 Exercises.- III. From Symmetry to the Borda Count and Other Procedures.- 3.1 Symmetry.- 3.1.1 Partial Orbits and Intensity of Comparisons.- 3.1.2 Neutrality.- 3.1.3 Reversal of Fortune.- 3.1.4 Reversal Geometry.- 3.1.5 Back to the Procedure Line.- 3.1.6 Reversal Bias Paradoxes.- 3.1.7 Borda Symmetry.- 3.1.8 Exercises.- 3.2 From Aggregating Pairwise Votes to the Borda Count.- 3.2.1 Borda and Aggregated Pairwise Votes.- 3.2.2 Geometric Representation.- 3.2.3 The Borda Dictionary.- 3.2.4 Borda Cross-Sections.- 3.2.5 The BC Cyclic Coordinates.- 3.2.6 The Borda Vector Space.- 3.2.7 Exercises.- 3.3 The Other Positional Voting Methods.- 3.3.1 What Can Accompany a F3 Tie Vote?.- 3.3.2 A Profile Coordinate Representation Approach.- 3.3.3 What Pairwise Outcomes Can Accompany a w s Tally?.- 3.3.4 Probability Computations.- 3.3.5 Exercises.- 3.4 Multiple Voting Schemes.- 3.4.1 From Multiple Methods to Approval Voting.- 3.4.2 No Good Deed Goes Unpunished.- 3.4.3 Comparisons.- 3.4.4 Averaged Multiple Voting Systems.- 3.4.5 Procedure Strips.- 3.4.6 Exercises.- 3.5 Other Election Procedures.- 3.5.1 Other Procedures.- 3.5.2 Ordinal Procedures.- 3.5.3 Scoring Runoffs.- 3.5.4 Comparisons of Positional Outcomes.- 3.5.5 Plurality or a Runoff?.- 3.5.6 Cardinal Procedures.- 3.5.7 Exercises.- IV. Many Profiles; Many New Paradoxes.- 4.1 Weak Consistency: The Sum of the Parts.- 4.1.1 Other Uses of Convexity.- 4.1.2 An L of an Agenda.- 4.1.3 Condorcet Extensions.- 4.1.4 Other Pairwise Procedures.- 4.1.5 Maybe "if's " and "and's", but no "or's" or "but's".- 4.1.6 A General Theorem.- 4.1.7 Exercises.- 4.2 From Involvement and Monotonicity to Manipulation.- 4.2.1 A Profile Angle.- 4.2.2 Positively Involved.- 4.2.3 Monotonicity.- 4.2.4 A General Theorem Using Profiles.- 4.2.5 Other Admissible Directions.- 4.2.6 Gibbard-Satterthwaite and Manipulable Procedures.- 4.2.7 Measuring Suspectibility to Manipulation.- 4.2.8 Exercises.- 4.3 Proportional Representation.- 4.3.1 Hare and Single Transferable Vote.- 4.3.2 The Apportionment Problem.- 4.3.3 Something Must Go Wrong - Alabama Paradox.- 4.3.4 A Better Improved Method?.- 4.3.5 More Surprises, but not Problems.- 4.3.6 House Monotone Methods.- 4.3.7 Unworkable Methods.- 4.3.8 Who Cares About Quota?.- 4.3.9 Big States, Small States.- 4.3.10 The Translation Bias.- 4.3.11 Sliding Bias.- 4.3.12 If the State of Washington Had Only 836 More People.- 4.3.13 A Solution.- 4.3.14 Exercises.- 4.4 Arrow's Theorem.- 4.4.1 A Sen Type Theorem.- 4.4.2 Universal Domain and IIA.- 4.4.3 Involvement and Voter Responsiveness.- 4.4.4 Arrow's Theorem.- 4.4.5 A Dictatorship or an Informational Problem?.- 4.4.6 Elementary Algebra.- 4.4.7 The $${F_{{c_i},{c_j}}}$$ Level Sets.- 4.4.8 Some Existence Theorems.- 4.4.9 Intensity IIA.- 4.4.10 Exercises.- 4.5 Characterizations of Scoring, Positional and Borda.- 4.5.1 Strong and Weak Consistency.- 4.5.2 Characterization of Scoring Rules.- 4.5.3 Positional Voting Methods.- 4.5.4 Axiomatic Characterizations of the BC.- 4.5.5 Generalized Positional Voting.- 4.5.6 Exercises.- Notes.- References.