Mathematics and Democracy
Designing Better Voting and Fair-Division Procedures
1st Edition
Published on 2. December 2009
392 pages
978-1-4008-3559-1 (ISBN)
Description
Voters today often desert a preferred candidate for a more viable second choice to avoid wasting their vote. Likewise, parties to a dispute often find themselves unable to agree on a fair division of contested goods. In Mathematics and Democracy, Steven Brams, a leading authority in the use of mathematics to design decision-making processes, shows how social-choice and game theory could make political and social institutions more democratic. Using mathematical analysis, he develops rigorous new procedures that enable voters to better express themselves and that allow disputants to divide goods more fairly.
One of the procedures that Brams proposes is "approval voting," which allows voters to vote for as many candidates as they like or consider acceptable. There is no ranking, and the candidate with the most votes wins. The voter no longer has to consider whether a vote for a preferred but less popular candidate might be wasted. In the same vein, Brams puts forward new, more equitable procedures for resolving disputes over divisible and indivisible goods.
One of the procedures that Brams proposes is "approval voting," which allows voters to vote for as many candidates as they like or consider acceptable. There is no ranking, and the candidate with the most votes wins. The voter no longer has to consider whether a vote for a preferred but less popular candidate might be wasted. In the same vein, Brams puts forward new, more equitable procedures for resolving disputes over divisible and indivisible goods.
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01/2008
Princeton University Press
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Person
Steven J. Brams is professor of politics at New York University. He is the author of Theory of Moves, 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.
Content
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:
Divide-the-Dollar 252
11.1. Introduction 252
11.2. DD1: A Reasonable Payoff Scheme 254
11.3. DD2: Adding a Second Stage 257
11.4. DD3: Combining DD1 and DD2 262
11.5. The Solutions with Entitlements 263
11.6. Summary and Conclusions 266
Appendix 267
Chapter 12: Allocating Multiple Homogeneous Divisible Goods:
Adjusted Winner 271
12.1. Introduction 271
12.2. Proportionality, Envy-Freeness, and Efficiency 272
12.3. Adjusted Winner (AW) 273
12.4. Issues at Camp David 275
12.5. The AW Solution 279
12.6. Practical Considerations 282
12.7. Summary and Conclusions 287
Chapter 13: Allocating a Single Heterogeneous Good:
Cutting a Cake 289
13.1. Introduction 289
13.2. Cut-and-Choose: An Example 290
13.3. The Surplus Procedure (SP) 292
13.4. Three or More Players: Equitability and Envy-Freeness May Be Incompatible 296
13.5. The Squeezing Procedure 297
13.6. The Equitability Procedure (EP) 299
13.7. Summary and Conclusions 303
Chapter 14: Allocating Divisible and Indivisible Goods 305
14.1. Introduction 305
14.2. Definitions and Assumptions 306
14.3. Difficulties with Equal and Proportional Reductions in the High Bids 308
14.4. The Gap Procedure 312
14.5. Pareto-Optimality 314
14.6. Envy-Freeness: An Impossible Dream 316
14.7. Sincerity and In dependence 322
14.8. Extending the Gap Procedure 323
14.9. Other Applications 324
14.10. Summary and Conclusions 327
Chapter 15: Summary and Conclusions 329
Glossary 337
References 343
Index 363
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:
Divide-the-Dollar 252
11.1. Introduction 252
11.2. DD1: A Reasonable Payoff Scheme 254
11.3. DD2: Adding a Second Stage 257
11.4. DD3: Combining DD1 and DD2 262
11.5. The Solutions with Entitlements 263
11.6. Summary and Conclusions 266
Appendix 267
Chapter 12: Allocating Multiple Homogeneous Divisible Goods:
Adjusted Winner 271
12.1. Introduction 271
12.2. Proportionality, Envy-Freeness, and Efficiency 272
12.3. Adjusted Winner (AW) 273
12.4. Issues at Camp David 275
12.5. The AW Solution 279
12.6. Practical Considerations 282
12.7. Summary and Conclusions 287
Chapter 13: Allocating a Single Heterogeneous Good:
Cutting a Cake 289
13.1. Introduction 289
13.2. Cut-and-Choose: An Example 290
13.3. The Surplus Procedure (SP) 292
13.4. Three or More Players: Equitability and Envy-Freeness May Be Incompatible 296
13.5. The Squeezing Procedure 297
13.6. The Equitability Procedure (EP) 299
13.7. Summary and Conclusions 303
Chapter 14: Allocating Divisible and Indivisible Goods 305
14.1. Introduction 305
14.2. Definitions and Assumptions 306
14.3. Difficulties with Equal and Proportional Reductions in the High Bids 308
14.4. The Gap Procedure 312
14.5. Pareto-Optimality 314
14.6. Envy-Freeness: An Impossible Dream 316
14.7. Sincerity and In dependence 322
14.8. Extending the Gap Procedure 323
14.9. Other Applications 324
14.10. Summary and Conclusions 327
Chapter 15: Summary and Conclusions 329
Glossary 337
References 343
Index 363
