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

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

by Steven J. Brams
     
 

ISBN-10: 0691133204

ISBN-13: 9780691133201

Pub. Date: 12/26/2007

Publisher: Princeton University Press

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…  See more details below

Overview

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.

About the Author:
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

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Product Details

ISBN-13:
9780691133201
Publisher:
Princeton University Press
Publication date:
12/26/2007
Pages:
390
Product dimensions:
6.30(w) x 9.30(h) x 1.20(d)

Table of Contents

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

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