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by Jonathan K. Hodge (Author), Richard E. Klima (Author)
About the Author
Jonathan K. Hodge: Grand Valley State University, Allendale, MI
Richard E. Klima: Appalachian State University, Boone, NC
About this book
The Mathematics of Voting and Elections: A Hands-On Approach, Second Edition, is an inquiry-based approach to the mathematics of politics and social choice. The aim of the book is to give readers who might not normally choose to engage with mathematics recreationally the chance to discover some interesting mathematical ideas from within a familiar context, and to see the applicability of mathematics to real-world situations. Through this process, readers should improve their critical thinking and problem solving skills, as well as broaden their views of what mathematics really is and how it can be used in unexpected ways. The book was written specifically for nonmathematical audiences and requires virtually no mathematical prerequisites beyond basic arithmetic. At the same time, the questions included are designed to challenge both mathematical and non-mathematical audiences alike. More than giving the right answers, this book asks the right questions. The book is fun to read, with examples that are not just thought-provoking, but also entertaining. It is written in a style that is casual without being condescending. But the discovery-based approach of the book also forces readers to play an active role in their learning, which should lead to a sense of ownership of the main ideas in the book. And while the book provides answers to some of the important questions in the field of mathematical voting theory, it also leads readers to discover new questions and ways to approach them. In addition to making small improvements in all the chapters, this second edition contains several new chapters. Of particular interest might be Chapter 12 which covers a host of topics related to gerrymandering.
Brief contents
Chapter 1. What’s So Good About Majority Rule? 1
The Mayor of Stickeyville 1
Anonymity, Neutrality, and Monotonicity 3
Majority Rule and May’s Theorem 5
Quota Systems 6
Back to May’s Theorem 8
Questions for Further Study 10
Answers to Starred Questions 12
Chapter 2. Le Pen, Nader, and Other Inconveniences 15
The Plurality Method 17
The Borda Count 18
Preference Orders 20
Back to Borda 22
May’s Theorem Revisited 23
Questions for Further Study 25
Answers to Starred Questions 30
Chapter 3. Back into the Ring 33
Condorcet Winners and Losers 35
Sequential Pairwise Voting 38
Instant Runoff 42
Putting It All Together 45
Questions for Further Study 46
Answers to Starred Questions 49
Chapter 4. Trouble in Democracy 53
Independence of Irrelevant Alternatives 54
Arrow’s Theorem 58
Pareto’s Unanimity Condition 63
Concluding Remarks 65
Questions for Further Study 65
Answers to Starred Questions 68
Chapter 5. Explaining the Impossible 71
Proving Arrow’s Theorem 72
Potential Solutions 79
Concluding Remarks 85
Questions for Further Study 86
Answers to Starred Questions 88
Chapter 6. Gaming the System 91
Strategic Voting 92
The Gibbard-Satterthwaite Theorem 93
Proving the Gibbard-Satterthwaite Theorem 95
Concluding Remarks 101
Questions for Further Study 102
Answers to Starred Questions 103
Chapter 7. One Person, One Vote? 105
Weighted Voting Systems 106
Dictators, Dummies, and Veto Power 109
Swap Robustness 110
Trade Robustness 113
Questions for Further Study 115
Answers to Starred Questions 118
Chapter 8. Calculating Corruption 121
The Banzhaf Power Index 122
The Shapley-Shubik Power Index 125
Banzhaf Power in Psykozia 128
A Splash of Combinatorics 130
Shapley-Shubik Power in Psykozia 133
Questions for Further Study 135
Answers to Starred Questions 138
Chapter 9. The Ultimate College Experience 143
The Electoral College 144
The Winner-Take-All Rule 146
Some History 148
in the Electoral College 149
Swing Votes and Perverse Outcomes 153
Alternatives to the Electoral College 157
Questions for Further Study 158
Answers to Starred Questions 162
Chapter 10. Trouble in Direct Democracy 163
Even More Trouble 165
The Separability Problem 166
Binary Preference Matrices 168
Testing for Separability 169
Some Potential Solutions 173
Questions for Further Study 179
Answers to Starred Questions 182
Chapter 11. Proportional (Mis)representation 185
The U.S. House of Representatives 186
Hamilton’s Apportionment Method 187
Jefferson’s Apportionment Method 190
Webster’s Apportionment Method 195
Three Apportionment Paradoxes 196
Hill’s Apportionment Method 198
Another Impossibility Theorem 200
Concluding Remarks 201
Questions for Further Study 202
Answers to Starred Questions 205
Chapter 12. Choosing Your Voters 207
Gerrymandering 209
Rules for Redistricting 214
Geometry and Compactness 215
Partisan Symmetry 218
The Efficiency Gap 221
Concluding Remarks 223
Questions for Further Study 224
Answers to Starred Questions 227
Bibliography 229
Index 233
Series: Mathematical World (Book 30)
Pages: 238 pages
Publisher: American Mathematical Society; 2 edition (October 1, 2018)
Language: English
ISBN-10: 1470442876
ISBN-13: 978-1470442873
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