Condorcet's Paradox / Edition 1

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Overview

The book is a compilation of the research that has been conducted in the study of Condorcet's Paradox for two centuries. It begins with a historical overview of the discovery of Condorcet's Paradox in the 18th Century, provides an exhaustive survey of numerous empirical studies that have been conducted to find actual occurrences of the paradox, and compiles all of the research that has been done to develop mathematical representations for the probability that the paradox will be observed. The book is unique since it focuses attention strictly on this one voting paradox, and it combines all of the different approaches that have been used to study this very interesting phenomenon.

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Product Details

  • ISBN-13: 9783540337980
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 8/29/2006
  • Series: Theory and Decision Library C Series , #40
  • Edition description: 2006
  • Edition number: 1
  • Pages: 292
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.75 (d)

Table of Contents


Elections and Voting Paradoxes     1
Introduction     1
The Case of More than Two Candidates     4
Borda's Paradox     7
Actual Occurrences of Various Forms of Borda's Paradox     8
Borda's Solution to the Possibility of Borda's Paradox     9
A Characterization of Borda Rule     15
Condorcet's Paradox     16
A Characterization of PMR     19
Condorcet's Other Paradox     21
The Paradox of Multiple Elections     24
The Vote Trading Paradox     25
The No Show Paradox     27
Other Voting Paradoxes     28
Conclusion     29
Condorcet's Paradox     31
Introduction     31
Riker's Empirical Studies     33
Other Empirical Studies     37
Monte-Carlo Simulation Studies     48
Conditions that Prohibit Condorcet's Paradox     51
Variations of Condorcet's Paradox     56
Conclusion     58
The Cases of Two and Three Candidates     59
Introduction     59
The Problem with Three Candidates     61
Probabilities with Balanced Preferences     66
Expected Balance forVoting Situations     69
A Fixed Number of Voters     69
A Variable Number of Voters     75
Expected Balance for Individual Preferences     81
Dual Culture Condition     83
Impartial Culture Condition     88
Uniform Culture Condition     92
Other IC-IAC Connections     94
The Impact of Unbalanced Preferences     99
Other Representations     101
Conclusion     104
The Case of More than Three Candidates     107
Introduction     107
Representations for Four-Candidate Elections     107
A PMRW Exists with Four Candidates     107
PMR is Transitive with Four Candidates     113
Probabilities for Four Candidates with Large Electorates     116
More than Four Candidates     117
Complete Breakdown by PMR     117
General PMR Relationships for More than Four Candidates     120
Enumerated Values of P[Characters not reproducible][subscript MRW](m,n,IC)     125
PMRW Probability Representations with Small m     126
PMRW Probability Representations with Small n     130
Limiting Probabilities with More than Four Candidates     138
Other Related Results      139
Spatial Models     140
Supermajority Rules     141
Condorcet Committees     144
Linear Extension Majority Cycles     150
Geometric Models     151
Conclusion     152
The Impact of Societal Factors     153
Introduction     153
Population Specific Measures of Homogeneity     155
Non-Comparison Population Measures     156
Comparison Population Measures     159
Situation Specific Measures of Homogeneity     160
The Effectiveness of Measures of Social Homogeneity     165
Requiring More Coherence in Voters' Preferences     168
Voter Abstention     171
Degrees of Voter Indifference between Candidates     183
The Impact of Intransitive Voter Preferences     190
The Impact of Uncertainty     193
Conclusion     194
The Impact of Coherent Preferences     195
Introduction     195
Methods for Obtaining Representations     195
EUPIA     197
EUPIA2     201
Proximity to Single-Troughed Preferences     212
The Impact of Polarizing Candidates     213
Limiting Distributions for Large Electorates     217
Cumulative Probabilities that a PMRW Exists     219
Proportions of Profiles with Specified Parameters     221
The Impact of an Overall Unifying Candidate     224
The Impact of Ward's Condition     226
Ehrhart Polynomials     228
Conclusion     228
Individual Intransitivity     229
Introduction     229
Algebraic and Probabilistic Choosers     230
May's Model     231
Probabilistic Chooser Models     234
Algebraic and Probabilistic Chooser Coincidence     235
Weak Maximal Alternatives     240
Attribute Independence - Discriminatory Power     244
The Impact of Single-Peakedness     256
Strict Maximal-Minimal Reversal     258
Other Related Representations     260
Conclusion     263
References     265
Index     285
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