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

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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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Editorial Reviews

Plus Magazine
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!
Science - Iain McLean
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.
SIAM News - Philip J. Davis
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.
Mathematical Reviews - D. Marc Kilgour
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.
Political Science Quarterly - Marek Kaminski
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.
+"Plus Magazine ianne Freiberger

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!
Time Magazines Higher Education
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.
From the Publisher

"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

Times Higher Education
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.
Science
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
SIAM News
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
Mathematical Reviews
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
Political Science Quarterly
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
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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)

Meet the Author

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".

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Read an Excerpt

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).

1.1. INTRODUCTION

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.

1.2. BACKGROUND

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.

1.3. EARLY HISTORY

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.

(Continues...)



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.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
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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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