Condorcet's Paradox

1. Auflage 2006
ISBN: 978-3-540-33799-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 292 Seiten

Condorcet’s Paradox has been formally studied by an amazing number of people in many different contexts for more than two centuries. Peter Fishburn introduced the basic notion of the Paradox to me in 1971 during a course in Social Choice Theory at Pennsylvania State University. My immediate response to seeing the simple example that he presented was that this phenomenon certainly could not be very likely to ever be observed in reality. Peter quickly suggested that I should work on developing some representations for the probability that the Paradox might occur, and very soon thereafter that pursuit began. It is only after 35 years of effort, with a lot of help from Peter, that I now feel that a good answer can be given to the challenge that was presented in that classroom in 1971. Many people have suggested to me over the years that a book like this should be completed, since the source material is spread over such a wide variety of disciplines of a- demic journals and books that it is very difficult for people to know what has been done, and has not been done, in this area of determining representations for the probability that Condorcet’s Paradox would ever be observed in reality.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;1 Elections and Voting Paradoxes;11
3.1;1.1 Introduction;11
3.2;1.2 The Case of More than Two Candidates;14
3.3;1.3 Borda’s Paradox;17
3.4;1.4 Condorcet’s Paradox;26
3.5;1.5. Condorcet’s Other Paradox;31
3.6;1.6 The Paradox of Multiple Elections;34
3.7;1.7 The Vote Trading Paradox;35
3.8;1.8 The No Show Paradox;37
3.9;1.9 Other Voting Paradoxes;38
3.10;1.10 Conclusion;39
4;2 Condorcet’s Paradox;40
4.1;2.1 Introduction;40
4.2;2.2 Riker’s Empirical Studies;42
4.3;2.3 Other Empirical Studies;46
4.4;2.4 Monte-Carlo Simulation Studies;57
4.5;2.5 Conditions that Prohibit Condorcet’s Paradox;60
4.6;2.6 Variations of Condorcet’s Paradox;65
4.7;2.7 Conclusion;67
5;3 The Cases of Two and Three Candidates;68
5.1;3.1 Introduction;68
5.2;3.2 The Problem with Three Candidates;70
5.3;3.3 Probabilities with Balanced Preferences;75
5.4;3.4 Expected Balance for Voting Situations;78
5.5;3.5 Expected Balance for Individual Preferences;90
5.6;3.6 Uniform Culture Condition;101
5.7;3.7 Other IC-IAC Connections;103
5.8;3.8 The Impact of Unbalanced Preferences;108
5.9;3.9 Other Representations;110
5.10;3.10 Conclusion;113
6;4 The Case of More than Three Candidates;115
6.1;4.1 Introduction;115
6.2;4.2 Representations for Four-Candidate Elections;115
6.3;4.3 More than Four Candidates;125
6.4;4.4 Other Related Results;147
6.5;4.5 Conclusion;160
7;5 The Impact of Societal Factors;161
7.1;5.1 Introduction;161
7.2;5.2 Population Specific Measures of Homogeneity;163
7.3;5.3 Situation Specific Measures of Homogeneity;168
7.4;5.4 The Effectiveness of Measures of Social Homogeneity;173
7.5;5.5 Requiring More Coherence in Voters’ Preferences;176
7.6;5.6 Voter Abstention;179
7.7;5.7 Degrees of Voter Indifference between Candidates;191
7.8;5.8 The Impact of Intransitive Voter Preferences;198
7.9;5.9 The Impact of Uncertainty;201
7.10;5.10 Conclusion;202
8;6 The Impact of Coherent Preferences;203
8.1;6.1 Introduction;203
8.2;6.2 Methods for Obtaining Representations;203
8.3;6.3 Proximity to Single-Troughed Preferences;220
8.4;6.4 The Impact of Polarizing Candidates;221
8.5;6.5 Limiting Distributions for Large Electorates;225
8.6;6.6 Cumulative Probabilities that a PMRW Exists;227
8.7;6.7 Proportions of Profiles with Specified Parameters;229
8.8;6.8 The Impact of an Overall Unifying Candidate;232
8.9;6.9 The Impact of Ward’s Condition;234
8.10;6.10 Ehrhart Polynomials;236
8.11;6.11 Conclusion;236
9;7 Individual Intransitivity;237
9.1;7.1 Introduction;237
9.2;7.2 Algebraic and Probabilistic Choosers;238
9.3;7.3 May’s Model;239
9.4;7.4 Probabilistic Chooser Models;242
9.5;7.5 Algebraic and Probabilistic Chooser Coincidence;243
9.6;7.6 Weak Maximal Alternatives;248
9.7;7.7 Attribute Independence - Discriminatory Power;252
9.8;7.8 The Impact of Single-Peakedness;264
9.9;7.9 Strict Maximal-Minimal Reversal;266
9.10;7.10 Other Related Representations;268
9.11;7.11 Conclusion;271
10;References;272
11;Index;292