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

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

by Steven J. Brams
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ISBN-13: 9780691133218
Publisher: Princeton University Press
Publication date: 01/06/2008
Edition description: New Edition
Pages: 392
Product dimensions: 6.00(w) x 9.25(h) x (d)

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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Mathematics and Democracy Designing Better Voting and Fair-Division Procedures

By Steven J. Brams Princeton University Press
Copyright © 2008
Princeton University Press
All right reserved.

ISBN: 978-0-691-13321-8

Chapter One Electing a Single Winner: Approval Voting in Practice

Note: This chapter is adapted from Brams and Fishburn (2005) with permission of Springer Science and Business Media; see also Brams (2002, 2006b) and Brams and Fishburn (1992a).


It may come as a surprise to some that there is a science of elections, whose provenance can be traced back to the Marquis de Condorcet in eighteenth-century France, Charles Dodgson (Lewis Carroll) in nineteenth-century England, and Kenneth Arrow in twentieth-century America. Since Arrow published his seminal book, Social Choice and Individual Values, more than fifty years ago (Arrow, 1951, 1963)-for which in large part he received the Nobel Memorial Prize in Economics in 1972-there have been thousands of articles and hundreds of books published on everything from the mathematical properties of voting systems to empirical tests of the propensity of different systems to elect centrist candidates.

The 2000 U.S. presidential election highlighted, among other things, the frailties of voting machines and the seeming arbitrariness of such venerable U.S. institutions as the Electoral College and the Supreme Court. Political commentary has focused on these aspects but given very littleattention to alternative voting systems, about which the science of elections has much to say.

Several alternative systems for electing a single winner have been shown to be far superior to plurality voting (PV)-the most common voting system used in the United States as well as in many other places-in terms of a number of criteria. PV, which allows citizens to vote for only one candidate, suffers from a dismaying flaw. In any race with more than two candidates, PV may elect the candidate least acceptable to the majority of voters. This frequently happens in a three-way contest, when the majority splits its votes between two centrist candidates, enabling a candidate on the left or right to defeat both centrists. PV also forces minor-party candidates into the role of spoilers, as was demonstrated in the 2000 presidential election with the candidacy of Ralph Nader. Nader received only 2.7 percent of the popular vote, but this percentage was decisive in an extremely close contest between the two major-party candidates.

Of the alternatives to PV, I recommend approval voting (AV), on both practical and theoretical grounds, in single-winner elections. Proposed independently by several analysts in the 1970s (Brams and Fishburn, 1983, 2007, ch. 1), AV is a voting procedure in which voters can vote for, or approve of, as many candidates as they wish in multicandidate elections-that is, elections with more than two candidates. Each approved candidate receives one vote, and the candidate with the most votes wins.

The candidate with the most votes need not win in an election. Merrill and Nagel (1987) make the useful distinction between a balloting method, which describes how voters can legally vote (e.g., for one candidate or for more than one), and a decision rule that determines a winner (e.g., the candidate with a plurality wins, or the candidate preferred to all others in pairwise comparisons wins). For convenience, I use the shorthand of AV to mean approval balloting with a plurality decision rule, but I consider other ways of aggregating approval votes later.

In the United States, the case for AV seems particularly strong in primary and nonpartisan elections, which often draw large fields of candidates. Here are some commonsensical arguments for AV that have been made:

1. It gives voters more flexible options. They can do exactly what they can under PV-vote for a single favorite-but if they have no strong preference for one candidate, they can express this fact by voting for all candidates they find acceptable. In addition, if a voter's most preferred candidate has little chance of winning, that voter can vote for both a first choice and a more viable candidate without worrying about wasting his or her vote on the less popular candidate.

2. It helps elect the strongest candidate. Today the candidate supported by the largest minority often wins, or at least makes the runoff if there is one. Under AV, by contrast, the candidate with the greatest overall support will generally win. In particular, Condorcet winners, who can defeat every other candidate in separate pairwise contests, almost always win under AV, whereas under PV they often lose because they split the vote with one or more other centrist candidates.

3. It will reduce negative campaigning. AV induces candidates to try to mirror the views of a majority of voters, not just cater to minorities whose voters could give them a slight edge in a crowded plurality contest. It is thus likely to cut down on negative campaigning, because candidates will have an incentive to try to broaden their appeals by reaching out for approval to voters who might have a different first choice. Lambasting such a choice would risk alienating this candidate's supporters and losing their approval.

4. It will increase voter turnout. By being better able to express their preferences, voters are more likely to vote in the first place. Voters who think they might be wasting their votes, or who cannot decide which of several candidates best represents their views, will not have to despair about making a choice. By not being forced to make a single-perhaps arbitrary-choice, they will feel that the election system allows them to be more honest, which will make voting more meaningful and encourage greater participation in elections.

5. It will give minority candidates their proper due. Minority candidates will not suffer under AV: their supporters will not be torn away simply because there is another candidate who, though less appealing to them, is generally considered a stronger contender. Because AV allows these supporters to vote for both candidates, they will not be tempted to desert the one who is weak in the polls, as under PV. Hence, minority candidates will receive their true level of support under AV, even if they cannot win. This will make election returns a better reflection of the overall acceptability of candidates, relatively undistorted by insincere or strategic voting, which is important information often denied to voters today.

6. It is eminently practicable. Unlike more complicated ranking systems, which suffer from a variety of theoretical as well as practical defects, AV is simple for voters to understand and use. Although more votes must be tallied under AV than under PV, AV can readily be implemented on existing voting machines. Because AV does not violate any state constitutions in the United States (or, for that matter, the constitutions of most countries in the world), it requires only an ordinary statute to enact.

Voting systems that involve ranking candidates may appear, at first blush, more appealing than AV. One, the Borda count, awards points to candidates according to their ranking. Another is the Hare system of single transferable vote (STV)-with variants called the "alternative vote" and "instant runoff"- in which candidates receiving the fewest first-choice votes are progressively eliminated. Their votes are transferred to second choices-and lower choices if necessary-until one candidate emerges with a majority of voters.

Compared with AV, these systems have serious drawbacks. The Borda count fosters "insincere voting" (for example, ranking a second choice at the bottom if that candidate is considered the strongest threat to one's top choice) and is also vulnerable to "irrelevant candidates" who cannot win but can affect the outcome. STV may eliminate a centrist candidate early and thereby elect one less acceptable to the majority. It also suffers from "nonmonotonicity," in which voters, by raising the ranking of a candidate, may actually cause that candidate to lose-just the opposite of what one would want to happen. I give examples of these drawbacks in the appendix to chapter 2.

As cherished a principle as "one person, one vote" is in single-winner elections, democracies, I believe, can benefit more from the alternative principle of "one candidate, one vote," whereby voters make judgments about whether each candidate on the ballot is acceptable or not. The latter principle makes the tie-in of a vote not to the voter but rather to the candidates, which is arguably more egalitarian than artificially restricting voters to casting only one vote in multicandidate races. This principle also affords voters an opportunity to express their intensities of preference by approving of, for example, all candidates except one they might despise.

Although AV encourages sincere voting, it does not altogether eliminate strategic calculations. Because approval of a less-preferred candidate can hurt a more-preferred approved candidate, the voter is still faced with the decision of where to draw the line between acceptable and unacceptable candidates. A rational voter will vote for a second choice if his or her first choice appears to be a long shot-as indicated, for example, by polls-but the voter's calculus and its effects on outcomes is not yet well understood for either AV or other voting procedures.

While AV is a strikingly simple election reform for finding consensus choices in single-winner elections, in elections with more than one winner- such as for a council or a legislature-AV would not be desirable if the goal is to mirror a diversity of views, especially of minorities; for this purpose, other voting systems should be considered, as I will discuss in later chapters.

On the other hand, minorities may derive indirect benefit from AV in single-winner elections, because mainstream candidates, in order to win, will be forced to reach out to minority voters for the approval they (the mainstream candidates) need in order to win. Put another way, these candidates must seek the consent of minority voters to be the most approved, or consensus, choices. While promoting majoritarian candidates, therefore, AV induces them to be responsive to minority views.


In this chapter, I describe some uses of AV, which began in the thirteenth century. However, I concentrate on more recent adoptions of AV, beginning in 1987, by several scientific and engineering societies, including the

Mathematical Association of America (MAA), with about 32,000 members

American Mathematical Society (AMS), with about 30,000 members

Institute for Operations Research and Management Sciences (INFORMS), with about 12,000 members

American Statistical Association (ASA), with about 15,000 members

Institute of Electrical and Electronics Engineers (IEEE), with about 377,000 members

Smaller societies that use AV include, among others, the Public Choice Society, the Society for Judgment and Decision Making, the Social Choice and Welfare Society, the International Joint Conference on Artificial Intelligence, the European Association for Logic, Language and Information, and the Game Theory Society (see chapter 5).

Additionally, the Econometric Society has used AV (with certain emendations) to elect fellows since 1980 (Gordon, 1981); likewise, since 1981 the selection of members of the National Academy of Sciences (1981) at the final stage of balloting has been based on AV. Coupled with many colleges and universities that now use AV-from the departmental level to the schoolwide level-it is no exaggeration to say that several hundred thousand individuals have had direct experience with AV.

Probably the best-known official elected by AV today is the secretary-general of the United Nations (Brams and Fishburn, 1983). AV has also been used in internal elections by the political parties in some states, such as Pennsylvania, where a presidential straw poll using AV was conducted by the Democratic State Committee in 1983 (Nagel, 1984).

Bills to implement AV have been introduced in several state legislatures (see section 1.2). In 1987, a bill to enact AV in certain statewide elections passed the Senate but not the House in North Dakota. In 1990, Oregon used AV in a statewide advisory referendum on school financing, which presented voters with five different options and allowed them to vote for as many as they wished (Wright, 1990).

In the late 1980s, AV was used in some competitive elections in countries in Eastern Europe and the Soviet Union, where it was effectively "disapproval voting," because voters were permitted to cross off names on ballots but not to vote for candidates (Shabad, 1987; Keller, 1987, 1988; White, 1989; Federal Election Commission, 1989). But this procedure is logically equivalent to AV. Candidates not crossed off are, in effect, approved of, although psychologically there is almost surely a difference between approving and disapproving of candidates.

With this information as background, I trace in section 1.3 my early involvement, and that of several associates, with AV. In section 1.4 I discuss how AV came to be adopted by the different societies.

In section 1.5 I report on empirical analyses of ballot data of some professional societies that adopted AV; they help to answer the question of when AV can make a difference in the outcome of an election. In section 1.6 I investigate the extent to which AV elects "lowest common denominators," which has concerned even supporters of AV. In section 1.7 I discuss whether voting is "ideological" under AV.

The confrontation between the theory underlying AV, which is rigorously developed in chapter 2, and practice offers some interesting lessons on "selling" new ideas. The rhetoric of AV supporters (I include myself), who have put forward the kinds of arguments outlined in section 1.1, has been opposed not only by those supporting extant systems like plurality voting (PV)-including incumbents elected under PV-but also by those with competing ideas, particularly proponents of other voting systems like the Borda count and the Hare system of single transferable vote.

I conclude that academics probably are not the best sales people for two reasons: (1) they lack the skills and resources, including time, to market their ideas, even when they are practicable; and (2) they squabble among themselves. Because few if any ideas in the social sciences are certifiably "right" under all circumstances, squabbles may well be grounded in serious intellectual differences. Often, however, they are not.


In 1976, I was attracted by the concept of "negative voting" (NV), proposed in a brief essay by Boehm (1976) that was passed on to me by the late Oskar Morgenstern. Under NV, voters can either vote for one candidate or against one candidate, but they cannot do both. Independently, Robert J. Weber had begun working on AV (he was apparently the first to coin the term "approval voting").

When Weber and I met in the summer of 1976 at a workshop at Cornell University under the direction of William F. Lucas, it quickly became apparent that NV and AV are equivalent when there are three candidates. Under both systems, a voter can vote for just one candidate. Under NV, a voter who votes against one candidate has the same effect as a voter who votes for the other two candidates under AV. And voting for all three candidates under AV has the same effect as abstaining under both systems.

When there are four candidates, however, AV enables a voter better to express his or her preferences. While voting against one candidate under NV has the same effect as voting for the other three candidates under AV, there is no equivalent under NV for voting for two of the four candidates. More generally, under NV a voter can do everything that he or she can do under AV, but not vice versa, so AV affords voters more opportunity to express themselves.

Weber and I wrote up our results separately, as did three other analysts who worked independently on AV in the 1970s (discussed in Brams and Fishburn, 1983, 2007; see also Weber, 1995). But the idea of AV did not spring forth, full-blown, only about thirty years ago; its origins go back many centuries. Indeed, AV was actually used, beginning in the thirteenth century, in both Venice (Lines, 1986) and papal elections (Colomer and McLean, 1998); it was also used in elections in nineteenth-century England (G. Cox, 1987), among other places.


Excerpted from Mathematics and Democracy by Steven J. Brams
Copyright © 2008 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

Preface xiii


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


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

What People are Saying About This

Hannu Nurmi

Professor Brams is one of the leading political scientists of our time, and one of the best-known authorities in the field of applied decision, game, and social choice theory. So, the level of expectations regarding his scholarly output is much higher than for most other authors. Yet, I believe this book surpasses that level.
Hannu Nurmi, University of Turku, Finland

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