Mathematics and Democracy : Designing Better Voting and Fair-Division Procedures

by Brams, Steven J.

• ISBN13: 9781400835591
• ISBN10: 1400835593
• Copyright: 2010-02-10
• Publisher: Princeton Univ Pr

Summary

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. InMathematics 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.

Table of Contents

Preface	p. xiii
Voting Procedures	p. 1
Electing a Single Winner: Approval Voting in Practice	p. 3
Introduction	p. 3
Background	p. 6
Early History	p. 8
The Adoption Decisions in the Societies	p. 10
Does AV Make a Difference?	p. 14
Does AV Elect the Lowest Common Denominator?	p. 16
Is Voting Ideological?	p. 18
Summary and Conclusions	p. 21
Electing a Single Winner: Approval Voting in Theory	p. 23
Introduction	p. 23
Preferences and Strategies under AV	p. 25
Election Outcomes under AV and Other Voting Systems	p. 26
Stability of Election Outcomes	p. 37
Summary and Conclusions	p. 42
Appendix	p. 43
Electing a Single Winner: Combining Approval and Preference	p. 46
Introduction	p. 46
Definitions and Assumptions	p. 48
Preference Approval Voting (PAV)	p. 49
Fallback Voting (FV)	p. 52
Monotonicity of PAV and FV	p. 56
Nash Equilibria under PAV and FV	p. 58
The Effects of Polls in 3-Candidate Elections	p. 61
Summary and Conclusions	p. 66
Electing Multiple Winners: Constrained Approval Voting	p. 69
Introduction	p. 69
Background	p. 70
Controlled Roundings	p. 72
Further Narrowing: The Search May Be Futile	p. 75
Constrained Approval Voting (CAV)	p. 80
Unconstraining Votes: Two Alternatives to CAV	p. 82
Summary and Conclusions	p. 87
Electing Multiple Winners: The Minimax Procedure	p. 89
Introduction	p. 89
Minisum and Minimax Outcomes	p. 91
Minimax versus Minisum Outcomes: They May Be Antipodes	p. 97
Endogenous versus Restricted Outcomes	p. 101
Manipulability	p. 103
The Game Theory Society Election	p. 105
Summary and Conclusions	p. 108
Appendix	p. 109
Electing Multiple Winners:
Minimizing Misrepresentation	p. 112
Introduction	p. 112
Obstacles to the Implementation of Proportional Representation (PR)	p. 113
Integer Programming	p. 115
Monroe's System	p. 116
Assigning More than One Candidate to a Voter	p. 119
Approval Voting	p. 121
Fractional Assignments	p. 123
Noninteger k	p. 125
The Chamberlin-Courant System	p. 126
Tullock's System	p. 127
Weighted Voting	p. 129
Nonmanipulability	p. 130
Representativeness	p. 131
Hierarchical PR	p. 133
Summary and Conclusions	p. 136
Appendixes	p. 138
Selecting Winners in Multiple Elections	p. 143
Introduction	p. 143
Referendum Voting: An Illustration of the Paradox of Multiple Elections	p. 145
The Coherence of Support for Winning Combinations	p. 149
Empirical Cases	p. 155
Relationship to the Condorcet Paradox	p. 160
Normative Questions and Democratic Political Theory	p. 165
Yes-No Voting	p. 167
Summary and Conclusions	p. 169
FAIR-DIVISION PROCEDURES	p. 171
Selecting a Governing Coalition in a Parliament	p. 173
Introduction	p. 173
Notation and Definitions	p. 176
The Fallback (FB) and Build-Up (BU) Processes	p. 177
The Manipulability of FB and BU	p. 181
Properties of Stable Coalitions	p. 182
The Probability of Stable Coalitions	p. 186
The Formation of Majorities in the U.S. Supreme Court	p. 189
Summary and Conclusions	p. 193
Appendix	p. 195
Allocating Cabinet Ministries in a Parliament	p. 199
Introduction	p. 199
Apportionment Methods and Sequencing	p. 202
Sophisticated Choices	p. 206
The Twin Problems of Nonmonotonicity and Pareto-Nonoptimality	p. 209
Possible Solutions: Trading and Different Sequencing	p. 214
A 2-Party Mechanism	p. 215
Order of Choice and Equitability	p. 218
Summary and Conclusions	p. 220
Appendix	p. 221
Allocating Indivisible Goods: Help the Worst-Off or Avoid Envy?	p. 224
Introduction	p. 224
Maximin and Borda Maximin Allocations	p. 227
Characterization of Efficient Allocations	p. 229
Maximin and Borda Maximin Allocations May Be Envy-Ensuring	p. 234
Finding Envy-Unensuring Allocations	p. 244
Unequal Allocations and Statistics	p. 248
Summary and Conclusions	p. 250
Allocating a Single Homogeneous Divisible Good:
Divide-the-Dollar	p. 252
Introduction	p. 252
DD1: A Reasonable Payoff Scheme	p. 254
DD2: Adding a Second Stage	p. 257
DD3: Combining DD1 and DD2	p. 262
The Solutions with Entitlements	p. 263
Summary and Conclusions	p. 266
Appendix	p. 267
Allocating Multiple Homogeneous Divisible Goods:
Adjusted Winner	p. 271
Introduction	p. 271
Proportionality, Envy-Freeness, and Efficiency	p. 272
Adjusted Winner (AW)	p. 273
Issues at Camp David	p. 275
The AW Solution	p. 279
Practical Considerations	p. 282
Summary and Conclusions	p. 287
Allocating a Single Heterogeneous Good:
Cutting a Cake	p. 289
Introduction	p. 289
Cut-and-Choose: An Example	p. 290
The Surplus Procedure (SP)	p. 292
Three or More Players: Equitability and Envy-Freeness May Be Incompatible	p. 296
The Squeezing Procedure	p. 297
The Equitability Procedure (EP)	p. 299
Summary and Conclusions	p. 303
Allocating Divisible and Indivisible Goods	p. 305
Introduction	p. 305
Definitions and Assumptions	p. 306
Difficulties with Equal and Proportional Reductions in the High Bids	p. 308
The Gap Procedure	p. 312
Pareto-Optimality	p. 314
Envy-Freeness: An Impossible Dream	p. 316
Sincerity and In dependence	p. 322
Extending the Gap Procedure	p. 323
Other Applications	p. 324
Summary and Conclusions	p. 327
Summary and Conclusions	p. 329
Glossary	p. 337
References	p. 343
Index	p. 363
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