Disposing dictators, demystifying voting paradoxes

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1 Does Everything Go Wrong?......Page 18
What “Subtlety?”......Page 19
1.2 And the Proud Father Is…......Page 20
1.3 Enemies?......Page 24
White Dresses and Other Humiliations......Page 26
Enemies and Money......Page 27
Blocking Borda’s Publication......Page 29
1.4 Curse of Dimensionality......Page 30
1.5 Outline......Page 32
1.5.1 Dethroning Dictators, and Then Paradoxes......Page 33
1.5.2 The ``Will of the Voters'': What Is It?......Page 35
TWO Dethroning Dictators......Page 37
2.1.1 Arrow's Theorem......Page 38
2.1.2 Sen's Seminal Result......Page 40
2.1.3 Topological Dictators......Page 43
2.1.4 “Paradox of Voting'' and Condorcet's Triplets......Page 45
2.1.5 List's Lists......Page 46
2.1.6 Anscombe......Page 47
2.2 Commonality......Page 48
From Condorcet to List and Anscombe......Page 49
They Always Are Related......Page 50
Three or More Pairs......Page 52
The Role of Outliers......Page 55
2.2.2 Can We Trust the Majority Voting over Pairs?......Page 57
2.2.3 A Common Explanation......Page 59
2.3 Why Do These Negative Results Occur?......Page 60
2.3.1 Arrow's Theorem......Page 61
My Granddaughter and Arrow’s Dictator......Page 62
2.3.2 Sen's Result......Page 64
Creating All Possible Examples......Page 65
All Possible Illustrating Examples......Page 66
An Upset Child......Page 68
Is a Topological Dictator Misleading?......Page 69
Effectiveness......Page 71
Missing Information?......Page 73
Finding a Positive Result......Page 75
Is IIA a Reasonable Condition?......Page 76
IIA Is Everywhere!......Page 80
New Interpretations......Page 81
Resolutions?......Page 82
Society in Transition......Page 84
Sen’s Conditions Are Everywhere......Page 85
2.4.3 Topological Dictators and More Effective Agents......Page 87
2.5 Final Thoughts......Page 89
THREE Voting Dictionaries......Page 91
3.1 What Goes Wrong?......Page 93
3.1.2 Dictionaries......Page 94
3.1.3 Aggregation Rules......Page 97
3.2 Lassie and the Axiomatic Approach......Page 98
A Voting Example......Page 99
3.3 Dictionaries......Page 101
3.3.1 A Little Chaos......Page 103
Flip the Coin, Roll the Die!......Page 104
3.3.2 Chaos within Voting Theory......Page 107
3.3.3 Dictionary Listings for Positional Rules......Page 108
Good News?......Page 110
Any Real Benefit?......Page 112
3.4 Using the Dictionaries......Page 113
Practical Tools for Analysis......Page 114
3.4.1 Variety Coming from Varieties......Page 116
Four and More Candidates......Page 118
How Wild Can It Get?......Page 119
3.4.2 Other Dictionaries......Page 121
3.5.1 General Results......Page 124
3.5.2 Let Elementary Geometry Do the Work......Page 126
Plotting Points......Page 127
3.5.3 Procedure Lines......Page 128
More Candidates......Page 129
3.5.4 Other Rules, Such As Approval Voting......Page 130
3.5.5 A Working Tool for Actual Elections......Page 133
Tabarrok’s Analysis......Page 134
3.5.6 Back to the Original Problem: Creating Examples......Page 135
Finding Properties......Page 136
Creating Profiles from Procedure Lines......Page 138
FOUR Explaining All Voting Paradoxes......Page 140
4.1.1 Coffee Reflections......Page 141
4.1.2 The “Water'' for Voting Rules......Page 144
An Illustration......Page 145
Dimension Count and the Dimensionality Curse......Page 146
Tempering That Dimensionality Curse......Page 147
Basic Profiles, the Coffee Grounds for Voting......Page 150
Examples with Basic Profiles......Page 153
More Candidates, Another Basic Subspace......Page 154
4.1.4 A Creamy Addition: Positional Differences......Page 156
Using Reversal Profiles......Page 159
4.1.5 Anything Can Happen......Page 161
4.1.6 Sugar and Spice, and All Those Nice Cycles......Page 163
Electing Fred......Page 164
Ranking Wheel......Page 165
From Arrow to Tournaments to Engineering......Page 167
Ad Hoc Construction, or General Behavior?......Page 168
An Example......Page 171
All Possible Problems with the Borda Count......Page 172
4.1.7 Differences between Borda and Condorcet......Page 174
4.1.8 Kemeny, Dodgson, and Other Systems; Who Cares?......Page 175
Other Coordinate Systems......Page 177
4.2.1 Creating Examples......Page 178
4.2.2 Converting Molecules into Coffee, Sugar, and Cream Coordinates......Page 179
4.3 The Will of the Voters......Page 182
4.3.1 Selecting Conditions......Page 183
4.3.2 Identifying the Voters' Wishes......Page 185
4.4 Teaser about More Candidates......Page 187
4.4.1 Designing Profile Configurations......Page 188
4.4.2 An Interesting Relationship......Page 189
4.4.3 A New Approach......Page 192
Borda and Condorcet Coordinates......Page 193
4.5 Finding and Proving New Theorems......Page 195
4.5.2 Condorcet Winners and Losers......Page 197
4.5.3 Borda Winners and Losers......Page 198
4.5.4 Low-Hanging Fruit with n Candidates......Page 199
4.5.5 Borda versus Pairwise Rankings......Page 200
A Wealth of Expressions......Page 202
4.5.6 Borda versus Borda Rankings......Page 204
Have Fun!......Page 206
5.1 Our Standard Voting Rule......Page 207
5.1.1 The 2000 U.S. Presidential Election......Page 208
5.1.2 The 2002 French Presidential Election......Page 209
5.1.3 Reform, or Fighting Termites with Paint and Putty?......Page 212
5.1.4 Resolutions, but Other Problems......Page 214
5.2 Newton's Third Law of Politics......Page 215
5.2.1 Comparison with Pairwise Voting......Page 217
5.2.2 Stability and the Core......Page 219
Core versus Number of Issues......Page 221
The q-Rules......Page 222
5.2.3 McKelvey's Chaos Theorem......Page 223
5.3 Generic Stability of the Core......Page 225
5.4 More about Cycles and Chaos......Page 228
5.4.1 Condorcet n-Cycles......Page 229
5.4.2 Controlling Chaos......Page 230
5.5 Final Comments......Page 231
SIX Appendix Extending the Upset Child Example......Page 235
6.1 Source of the Problem......Page 236
6.2 Generalizations......Page 239
6.3 Level Sets......Page 243
References......Page 248
Index......Page 254