Basic Geometry of Voting

307 Pages·1995·21.595 MB·English

by  Professor Donald G. Saari (auth.)

Basic Geometry
of Voting
Springer-Ve rlag Berlin Heidelberg GmbH
Donald G. Saari
Basic Geometry
of Voting
With 102 Figures
Springer
Professor Donald G. Saari
Northwestern University
Department of Mathematics
2033 Sheridan Road
Evanston, IL 60208-2730, USA
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
SaarI, Donald G_:
Basic geometry of voting I Donald G. Saari. - Berlin:
Heidelberg : New York; Barcelona ; Budapest ; Hong Kong ;
London ; Milan ; Paris; Tokyo: Springer. 1995
ISBN 3-540-60064-7
ISBN 978-3-540-60064-0 ISBN 978-3-642-57748-2 (eBook)
DOI 10.1007/978-3-642-57748-2
This work is subject to copyright. AII rights are reserved, whether the whole ar part of the material is concerned,
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© Springer-Verlag Berlin Heidelberg 1995
Originally published by Physica-Verlag Heidelberg 1995
The use of registered names, trademarks, etc. in this publicati an does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant protective laws and regulations and therefore free for
general use.
SPIN 10507193 42/2202-543210 -Printed an acid-free paper
In memory of
Gene A. Saari
and for
Martha M. Saari
A dedicated team of idealistic, pragmatic radicals
who made a difference by improving the lives of many!
PREFACE
A surprise of this book is how the well-known intricacies and complexities of
elections can be identified and resolved with the comfortable geometry of our
three-dimensional world. This allows previously unpublished results and/or new
explanations for old assertions to be included. So, even though this book is
directed toward students and others interested in learning about the field, experts
will find much that is new. To help test understanding and to facilitate using
this material in a course, exercises follow each section.
As an introduction to what can go wrong, Chap. 1 catalogues the woes of
the Chair of a hypothetical academic department. (These examples are used
throughout the book.) The second section builds on this fable to raise issues;
this is followed by a selective history. The last section reminds the reader of
useful mathematical properties. The actual geometry starts in Chap. 2 where
standard terms are introduced with a geometric formulation.
The results of Chap. 3 directly contradict commonly accepted notions. Here it
is shown why the standard used by most of the field to judge procedures is highly
flawed. In this critical analysis of pairwise voting, foundations are developed to
analyze the debates central to this field since the 1780s. New approaches include
the "profile coordinate representations" and the "profile decomposition" which
make it almost trivial to construct examples illustrating all sorts of voting be
havior. The chapter ends by showing that Arrow's Theorem is not as disturbing
as commonly believed; instead, as I show, the conclusion is to be expected.
In Chap. 4, geometric methods are developed to analyze positional procedures
(generalizations of our standard plurality vote) which make it easy to understand
why different methods can lead to different outcomes. Indeed, deriving certain
new conclusions now is as simple as drawing lines on a triangle. As part of this
discussion, new sets of profile coordinates and decompositions are given which
allow us to quickly analyze almost any three-candidate example. For instance,
from this description we discover that pairwise and positional rankings can differ
because they rely on different kinds of information about the voters. Then the
fundamental properties of Borda Count are introduced.
While Chaps. 3 and 4 emphasize single profile issues, Chap. 5 describes those
fascinating scenarios involving several profiles. The first three sections, for exam
ple, show what can happen when subcommittees join, voters abstain or change
their opinions, or the procedure is manipulated. As a sampler, associated with
a new, elementary proof of the Gibbard-Satterthwaite Theorem is a technique
to analyze how susceptible a procedure is to being manipulated. In the last
Vll! PREFACE
two sections, which emphasizes proportional voting, new geometry techniques
explain the serious problems of apportionment methods. As most procedures
admit reasonable outcomes only through a fortuitous balance of compensating
errors, we must anticipate problems. Indeed, legal issues raised about the current
apportionment of Congressional seats for the USA, issues that reached the US
Supreme Court, are mentioned.
This book started as a "student" version of Geometry of Voting; I dropped
those sections primarily of interest to experts, reorganized the presentation to
represent how the material would be taught (the ordering of Geometry of Voting
is dictated by the mathematical development), and added exercises. However,
I could not resist adding several recent results. Thus, while portions of this
book overlaps material first developed in Geometry of Voting, it also contains
many previously unpublished conclusions. Some of these new approaches, for
instance, eliminate the two-century old challenge of analyzing the profiles from
the Condorcet - Borda debates of the 1780s.
When writing a book, one appreciates the critical importance of colleagues
and friends! This is particularly so when a computer program doesn't work and
I'm rescued by my colleague Clark Robinson. If you like the computer generated
pictures, thank Hollie Howard, a Northwestern University undergraduate who
worked with me for three delightful years; she programmed them with PiCTeX.
Thanks for encouragement, suggestions, corrections, comments, references, in
formation, critiques, etc., etc., go to Roko Aliprantis, Jean-Pierre Aubin, Steve
Brams, Len Evens, Eric Friedlander, Hollie Howard, Ehud Kalai, Jerry Kelly,
Vincent Merlin, Ken Mount, Diana Richards, Clark Robinson, Stan Reiter, Ka
tri Saari, Maurice Salles, Mark Satterthwaite, Carl Simon, Maria Tataru, Arnold
Urken, Steve Williams, our Thursday afternoon discussion group on the "math
ematics of the social sciences," and several others. Particular thanks go to my
former NSF project director Larry Rosenberg who, right up until he died, was
most supportive.
Donald G. Saari
Northwestern University
Evanston, Illinois
May, 1995
Typeset wi th AMS-TEX
CONTENTS
Chapter I. From an Election Fable to Election Procedures 1
1.1 An Electoral Fable 2
1.1.1 Time for the Dean 4
1.1.2 The Departmental Election 6
1.1. 3 Exercises 7
1.2 The Moral of the Tale 8
1.2.1 The Basic Goal 9
1.2.2 Other Political Issues 10
1.2.3 Strategic Behavior 11
1.2.4 Some Procedures Are Better than Others 12
1.2.5 Exercises 15
1.3 From Aristotle to "Fast Eddie" 15
1.3.1 Selecting a Pope 16
1.3.2 Procedure Versus Process 17
1.3.3 Jean-Charles Borda 18
1.3.4 Beyond Borda 20
1.4 What Kind of Geometry? 21
1.4.1 Convexity and Linear Mappings 22
1.4.2 Convex Hulls 25
1.4.3 Exercises 27
Chapter II. Geometry for Positional And Pairwise Voting 29
2.1 Ranking Regions 30
2.1.1 Normalized Election Tally 31
2.1.2 Ranking Regions 33
2.1.3 Exercises 35
2.2 Profiles and Election Mappings 38
2.2.1 The Election Mapping 40
2.2.2 The Geometry of Election Outcomes 42
2.2.3 Exercises 44
Chapter III. The Problem With Condorcet 45
3.1 Why Can't an Organization Be More Like a Person? 45
3.1.1 Confused, Irrational Voters 47
3.1.2 Information Lost from Pairwise Majority Voting 49
3.1.3 Reduced Profiles 51
3.1.4 Exercises 54
3.2 Geometry of Pairwise Voting 56
3.2.1 The Geometry of Cycles 60
3.2.2 Cyclic Profile Coordinates 63
3.2.3 Power of Cyclic Coordinates 65
x CONTENTS
3.2.4 The Return of Confused Voters 67
3.2.5 Exercises 7]
3.3 Black's Single-Peakedness 71
3.3.1 Black's Condition 72
3.3.2 Condorcet Winners and Losers 75
3.3.3 A Condorcet Improver~~nt 79
3.3.4 Exercises 82
3.4 Arrow's Theorem 83
3.4.1 A Sen Type Theorem 83
3.4.2 Universal Domain ane TT', 84
3.4.3 Involvement and Vot, r Re ponsiveness 85
3.4.4 Arrow's Theorem 87
3.4.5 A Dictatorship or an Informational Problem? 87
3.4.6 Elementary Algebra 89
3.4.7 The Fe c· Level Sets 92
3.4.8 Some i:ristence Theorems 94
3.4.9 Intensity IIA 96
3.4.10 Exercises 98
Chapter IV. Positional Voting And the Be 101
4.1 Positional Voting Methods 101
4.1.1 The Difference a Procedure Makes 102
4.1.2 An Equivalence Relationship for Voting Vectors 104
4.1.3 The Geometry of w s Outcomes 107
4.1.4 Exercises 108
4.2 What a Difference a Procedure Makes; Several Different Out-
comes 110
4.2.1 How Bad It Can Get 112
4.2.2 Properties of Sup(p) 113
4.2.3 The Procedure Line 115
4.2.4 Using the Procedure Line 118
4.2.5 Robustness of the Paradoxical Assertions 121
4.2.6 Proofs 122
4.2.7 Exercises 124
4.3 Positional Versus Pairwise Voting 125
4.3.1 Comparing Votes With a Fat Triangle 125
4.3.2 Positional Group Coordinates 127
4.3.3 Profile Sets 131
4.3.4 Some Comparisons 1:l6
4.3.5 Comparisons 137
4.3.6 How Varied Does It Get? 139
1.3.7 Exercises 139
4.4 Profile Decomposition 140
4.4.1 Neutrality and Reversal Bias 141
4.4.2 Reversal Sets 143
4.4.a Cancellation 145
4.4.4 Ha si c Profiles 118
4.4.5 Symmetry of Voting Vectors 150
4.4.6 Exercises 153
1.5 From Aggregating Pairwise Votes to the Borda Count 154
CONTENTS xi
4.5.1 Borda and Aggregated Pairwise Votes 156
4.5.2 Basic Profiles 159
4.5.3 Geometric Representation 163
4.5.4 The Borda Dictionary 165
4.5.5 Borda Cross-Sections 165
4.5.6 Exercises 167
4.6 The Other Positional Voting Methods 168
4.6.1 What Can Accompany a F3 Tie Vote? 170
4.6.2 A Profile Coordinate Representation Approach 172
4.6.3 What Pairwise Outcomes Can Accompany a w s Tally? 174
4.6.4 Probability Computations 177
4.6.5 Exercises 178
4.7 Multiple Voting Schemes 178
4.7.1 From Multiple Methods to Approval Voting 179
4.7.2 No Good Deed Goes Unpunished 182
4.7.3 Comparisons 185
4.7.4 Averaged Multiple Voting Systems 186
4.7.5 Procedure Strips 188
4.7.6 Exercises 189
4.8 Other Election Procedures 190
4.8.1 Other Pairwise Procedures 190
4.8.2 Runoffs 192
4.8.3 Scoring Runoffs 193
4.8.4 Comparisons of Positional Voting Outcomes 195
4.8.5 Plurality or a Runoff? 196
4.8.6 Exercises 197
Chapter V. Other Voting Issues 201
5.1 Weak Consistency: The Sum of the Parts 201
5.1.1 Other Uses of Convexity 204
5.1.2 An L of an Agenda 205
5.1.3 Condorcet Extensions 207
5.1.4 Other Pairwise Procedures 208
5.1.5 Maybe "If's" and "And's", But No "Or's" or "But's" 209
5.1.6 A General Theorem 212
5.1.7 Exercises 214
5.2 From Involvement and Monotonicity to Manipulation 215
5.2.1 Positively Involved 215
5.2.2 Monotonicity 219
5.2.3 A Profile Angle 224
5.2.4 A General Theorem Using Profiles 225
5.2.5 Other Admissible Directions 228
5.2.6 Exercises 230
5.3 Gibbard-Satterthwaite and Manipulable Procedures 231
5.3.1 Measuring Suspectibility to Manipulation 233
5.3.2 Exercises 242
5.4 Proportional Representation 242
5.4.1 Hare and Single Transferable Vote 244
5.4.2 The Apportionment Problem 246
5.4.3 Something Must Go Wrong - Alabama Paradox 250