Mathematics and Democracy

"Showing how social-choice theory and game theory could make political and social institutions more democratic, Brams uses mathematical analysis to develop new procedures that could enable voters to better express their preferences."--Times Higher Education "The image on the cover of Mathematics and Democracy shows four people pulling on two ropes. If they all pull, the knot will jam. The book's contents show, on the contrary, that sometimes mathematics and game theory can unjam the problems of voting."--Iain McLean, Science "In seven chapters, Brams proposes and dissects a range of, often very elegant, fair division procedures pertaining to different situations... Brams strengthens his arguments with a wealth of real-life examples, from US elections to the 1978 peace negotiations between Israel and Egypt. The mathematical results are amply illustrated with easy-to-follow examples... If you're interested in democracy, then this book makes eye-opening reading, and if you're planning on wielding power at some point in the future, then it should be compulsory!"--Marianne Freiberger, +Plus Magazine "Mathematics and Democracy is rich in analyses of historical cases... Read Mathematics and Democracy: You will learn of the vast number of voting options that have been mooted, and you will easily conclude that any proposed change, however minor, will arouse fury in some constituency somewhere."--Philip J. Davis, SIAM News "Increasingly, mathematicians are finding interesting problems in social science, a development that the previous books of Steven J. Brams helped to catalyze. Mathematics and Democracy, based on a selection of Brams's (mostly co-authored) papers, will add to his influence."--D. Marc Kilgour, Mathematical Reviews "Since the math is elementary and the problems familiar, the book can be read both by political scientists not allergic to formal reasoning and by amateurs of mathematics interested in politics. Voting practitioners and designers will be delighted to find thorough discussions of less-known methods. All of them will find the book an interesting introduction to the fascinating subfield of mathematically oriented political science that analyzes and invents constructive institutional solutions to social dilemmas."--Marek Kaminski, Political Science Quarterly

Steven J. Brams is professor of politics at New York University. He is the author of "Theory of Move"s, among many other books, and the coauthor of "The Win-Win Solution: Guaranteeing Fair Shares to Everybody" and "Fair Division: From Cake-Cutting to Dispute Resolution".

Preface xiii PART 1. VOTING PROCEDURES 1 Chapter 1: Electing a Single Winner: Approval Voting in Practice 3 1.1. Introduction 3 1.2. Background 6 1.3. Early History 8 1.4. The Adoption Decisions in the Societies 10 1.5. Does AV Make a Difference? 14 1.6. Does AV Elect the Lowest Common Denominator? 16 1.7. Is Voting Ideological? 18 1.8. Summary and Conclusions 21 Chapter 2: Electing a Single Winner: Approval Voting in Theory 23 2.1. Introduction 23 2.2. Preferences and Strategies under AV 25 2.3. Election Outcomes under AV and Other Voting Systems 26 2.4. Stability of Election Outcomes 37 2.5. Summary and Conclusions 42 Appendix 43 Chapter 3: Electing a Single Winner: Combining Approval and Preference 46 3.1. Introduction 46 3.2. Definitions and Assumptions 48 3.3. Preference Approval Voting (PAV) 49 3.4. Fallback Voting (FV) 52 3.5. Monotonicity of PAV and FV 56 3.6. Nash Equilibria under PAV and FV 58 3.7. The Effects of Polls in 3-Candidate Elections 61 3.8. Summary and Conclusions 66 Chapter 4: Electing Multiple Winners: Constrained Approval Voting 69 4.1. Introduction 69 4.2. Background 70 4.3. Controlled Roundings 72 4.4. Further Narrowing: The Search May Be Futile 75 4.5. Constrained Approval Voting (CAV) 80 4.6. Unconstraining Votes: Two Alternatives to CAV 82 4.7. Summary and Conclusions 87 Chapter 5: Electing Multiple Winners: The Minimax Procedure 89 5.1. Introduction 89 5.2. Minisum and Minimax Outcomes 91 5.3. Minimax versus Minisum Outcomes: They May Be Antipodes 97 5.4. Endogenous versus Restricted Outcomes 101 5.5. Manipulability 103 5.6. The Game Theory Society Election 105 5.7. Summary and Conclusions 108 Appendix 109 Chapter 6: Electing Multiple Winners: Minimizing Misrepresentation 112 6.1. Introduction 112 6.2. Obstacles to the Implementation of Proportional Representation (PR) 113 6.3. Integer Programming 115 6.4. Monroe's System 116 6.5. Assigning More than One Candidate to a Voter 119 6.6. Approval Voting 121 6.7. Fractional Assignments 123 6.8. Noninteger k 125 6.9. The Chamberlin-Courant System 126 6.10. Tullock's System 127 6.11. Weighted Voting 129 6.12. Nonmanipulability 130 6.13. Representativeness 131 6.14. Hierarchical PR 133 6.15. Summary and Conclusions 136 Appendixes 138 Chapter 7: Selecting Winners in Multiple Elections 143 7.1. Introduction 143 7.2. Referendum Voting: An Illustration of the Paradox of Multiple Elections 145 7.3. The Coherence of Support for Winning Combinations 149 7.4. Empirical Cases 155 7.5. Relationship to the Condorcet Paradox 160 7.6. Normative Questions and Democratic Political Theory 165 7.7. Yes-No Voting 167 7.8. Summary and Conclusions 169 PART 2. FAIR-DIVISION PROCEDURES 171 Chapter 8: Selecting a Governing Coalition in a Parliament 173 8.1. Introduction 173 8.2. Notation and Definitions 176 8.3. The Fallback (FB) and Build-Up (BU) Processes 177 8.4. The Manipulability of FB and BU 181 8.5. Properties of Stable Coalitions 182 8.6. The Probability of Stable Coalitions 186 8.7. The Formation of Majorities in the U.S. Supreme Court 189 8.8. Summary and Conclusions 193 Appendix 195 Chapter 9: Allocating Cabinet Ministries in a Parliament 199 9.1. Introduction 199 9.2. Apportionment Methods and Sequencing 202 9.3. Sophisticated Choices 206 9.4. The Twin Problems of Nonmonotonicity and Pareto-Nonoptimality 209 9.5. Possible Solutions: Trading and Different Sequencing 214 9.6. A 2-Party Mechanism 215 9.7. Order of Choice and Equitability 218 9.8. Summary and Conclusions 220 Appendix 221 Chapter 10: Allocating Indivisible Goods: Help the Worst-Off or Avoid Envy? 224 10.1. Introduction 224 10.2. Maximin and Borda Maximin Allocations 227 10.3. Characterization of Efficient Allocations 229 10.4. Maximin and Borda Maximin Allocations May Be Envy-Ensuring 234 10.5. Finding Envy-Unensuring Allocations 244 10.6. Unequal Allocations and Statistics 248 10.7. Summary and Conclusions 250 Chapter 11: Allocating a Single Homogeneous Divisible Good