ISBN: 3540337989
TITLE: Condorcets Paradox
AUTHOR: Gehrlein
TOC:

1 Elections and Voting Paradoxes 1
1.1 Introduction 1
1.2 The Case of More than Two Candidates 4
1.3 Borda's Paradox 7
1.3.1 Actual Occurrences of Various Forms of Borda's Paradox 8
1.3.2 Borda's Solution to the Possibility of Borda's Paradox 9
1.3.3 A Characterization of Borda Rule 15
1.4 Condorcet's Paradox 16
1.4.1 A Characterization of PMR 19
1.5. Condorcet's Other Paradox 21
1.6 The Paradox of Multiple Elections 24
1.7 The Vote Trading Paradox 25
1.8 The No Show Paradox 27
1.9 Other Voting Paradoxes 28
1.10 Conclusion 29
2 Condorcet's Paradox 31
2.1 Introduction 31
2.2 Riker's Empirical Studies 33
2.3 Other Empirical Studies 37
2.4 Monte-Carlo Simulation Studies 48
2.5 Conditions that Prohibit Condorcet's Paradox 51
2.6 Variations of Condorcet's Paradox 56
2.7 Conclusion 58
3 The Cases of Two and Three Candidates 59
3.1 Introduction 59
3.2 The Problem with Three Candidates 61
3.3 Probabilities with Balanced Preferences 66
3.4 Expected Balance for Voting Situations 69
 3.4.1 A Fixed Number of Voters 69
3.4.2 A Variable Number of Voters 75
3.5 Expected Balance for Individual Preferences 81
3.5.1 Dual Culture Condition 83
3.5.2 Impartial Culture Condition 88
3.6 Uniform Culture Condition 92
3.7 Other IC-IAC Connections 94
3.8 The Impact of Unbalanced Preferences 99
3.9 Other Representations 101
3.10 Conclusion I04
4 The Case of More than Three Candidates 107
4.I Introduction I07
4.2 Representations for Four-Candidate Elections 107
4.2.I A PMRW Exists with Four Candidates 107
4.2.2 PMR is Transitive with Four Candidates 113
4.2.3 Probabilities for Four Candidates with Large Electorates 116
4.3 More than Four Candidates 117
4.3.1 Complete Breakdown by PMR 117
4.3.2 General PMR Relationships for More than Four Candidates 120
4.3.3 Enumerated Values of PpMRW (m, n, IC) 125
4.3.4 PMRW Probability Representations with Small m 126
4.3.5 PMRW Probability Representations with Small n 130
4.3.6 Limiting Probabilities with More than Four Candidates 138
4.4 Other Related Results 139
4.4.1 Spatial Models 140
4.4.2 Supermajority Rules 141
4.4.3 Condorcet Committees 144
4.4.4 Linear Extension Majority Cycles 150
4.4.5 Geometrie Models 151
4.5 Conclusion 152
5 The Impact of Societal Factors 153
5.1 Introduction 153
5.2 Population Specific Measures of Homogeneity 155
5.2.1 Non-Comparison Population Measures 156
5.2.2 Comparison Population Measures 159
5.3 Situation Specific Measures of Homogeneity 160
5.4 The Effectiveness of Measures of Social Homogeneity 165
5.5 Requiring More Coherence in Voters' Preferences 168
5.6 Voter Abstention 171
5.7 Degrees of Voter Indifference between Candidates 183
5.8 The Impact of Intransitive Voter Preferences 190
5.9 The Impact of Uncertainty 193
5.10 Conclusion 194
6 The Impact of Coherent Preferences 195
6.1 Introduction 195
6.2 Methods for Obtaining Representations 195
6.2.1 EUPIA 197
6.2.2 EUPIA2 201
6.3 Proximity to Single-Troughed Preferences 212
6.4 The Impact of Polarizing Candidates 213
6.5 Limiting Distributions for Large Electorates 217
6.6 Cumulative Probabilities that a PMRW Exists 219
6.7 Proportions of Profiles with Specified Parameters 221
6.8 The Impact of an Overall Unifying Candidate 224
6.9 The Impact of Ward's Condition 226
6.10 Ehrhart Polynomials 228
6.11 Conclusion 228
7 Individual Intransitivity 229
7.1 Introduction 229
7.2 Algebraic and Probabilistic Choosers 230
7.3 May's Model 231
7.4 Probabilistic Chooser Models 234
7.5 Algebraic and Probabilistic Chooser Coincidence 235
7.6 Weak Maximal Alternatives 240
7.7 Attribute Independence - Discriminatory Power 244
7.8 The Impact of Single-Peakedness 256
7.9 Strict Maximal-Minimal Reversal 258
7.10 Other Related Representations 260
7.11 Conclusion 263
References 265
Index 285
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