Voting Paradoxes and Group Coherence

Transcription

1 William V. Gehrlein Dominique Lepelley Voting Paradoxes and Group Coherence The Condorcet Efficiency of Voting Rules 4y Springer

2 Contents 1 Voting Paradoxes and Their Probabilities Introduction The Case of More than Two Candidates Voting Paradoxes Incompatibility Paradoxes Monotonicity Paradoxes Choice Set Variance Paradoxes Empirical Evidence of the Existence of Voting Paradoxes Empirical Evidence of Condorcet's Paradox Empirical Evidence of Borda's Paradox Probability Representations for Voting Paradoxes Multinomial Probability Models for Voter Profiles Multinomial Probability Models - Limiting Case for n Probability Models for Voting Situations - Algebraic Approach Probability Models for Voting Situations - EUPIA Probability Models - Ehrhart Polynomials Probability Models - Barvinok's Algorithm Relevance of DC, IC, IAC and MC Based Probabilities General Arguments Results from the DC Assumption Results from IC-IAC Comparisons Homogeneity and Dependence Connections Conclusion 47 2 Condorcet's Paradox and Group Coherence Introduction Population Specific Measures of Homogeneity Situation Specific Measures of Homogeneity Weak Measures of Group Coherence 54

3 x Contents Strong Measures of Group Coherence Obtaining Probability Representations EUPIA Cumulative Probabilities that a PMRW Exists Proportions of Profiles with Specified Parameters Results with Strong Measures of Group Coherence Conclusion 78 3 Other Incompatibility Paradoxes Introduction Borda's Paradox : The Probability of Observing a Strict Borda Paradox The Probability of Observing a Strong Borda Paradox Overall Probabilities for Borda's Paradox Condorcet's Other Paradox A More Relaxed Condition Another Condition Conclusion Other Voting Paradoxes Choice Set Variance Paradoxes Ostrogorski's Paradox The Majority Paradox Monotonicity Paradoxes Monotonicity Paradox Probabilities No Show Paradox Probabilities The Instability Paradox Conclusion Condorcet Efficiency and Social Homogeneity Introduction The Desirability of Using Simple Voting Rules Early Research on the Condorcet Efficiency of Voting Rules Early Numerical Analysis of Condorcet Efficiency Probability Representations for Condorcet Efficiency Summary of Condorcet Efficiency Results The Impact of Social Homogeneity on Efficiency Population Specific Measures of Homogeneity Situation Specific Measures of Homogeneity Strong Condorcet Efficiency of Voting Rules Spatial Modeling Results Summary of Social Homogeneity Results 198

4 Contents xi 6 Coherence and the Efficiency Hypothesis Introduction Numerical Evidence Condorcet Efficiency with Single Peaked Preferences Efficiency with Weak Measures of Group Coherence Condorcet Efficiency of PR with Weak Measures Condorcet Efficiency of NPR with Weak Measures Condorcet Efficiency of BR with Weak Measures Single-Stage Voting Rules with Weak Measures Single-Stage Voting Rules: A Borda Compromise Two-Stage Rule Efficiencies with Weak Measures Efficiency with Strong Measures of Group Coherence Single-Stage Rule Representations with Strong Measures Single-Stage Rule Efficiencies with Strong Measures Two-Stage Rule Representations with Strong Measures Two-Stage Rule Efficiencies with Strong Measures Conclusion Other Characteristics of Voting Rules Introduction Empirical Studies of Condorcet Efficiency Single-Stage Voting Rules Two-Stage Voting Rules Practical Factors and Condorcet Efficiency Voter Indifference and Condorcet Efficiency The Forced Ranking Option Modifying WSR's for Voter Indifference Voter Abstention and Condorcet Efficiency Two-Candidate Elections Three-Candidate Elections The Presence of a PMR Cycle and Condorcet Efficiency Methods for Breaking PMR Cycles The Efficiency of WSR's when PMR Cycles Exist The Impact of Removing Candidates Results from Saari's Analysis of WSR's Characterizations of BR Potential for Manipulation Empirical Results on BR Manipulability Analytical Studies of BR Manipulation Conclusion The Significance of Voting Rule Selection Introduction Same Winner with Two Voting Rules 295

5 xii Contents Two Voting Rules Winner Coincidence Two Voting Rules Winner Coincidence with the PMRW The Probability That All WSR's Elect the Same Winner Homogeneity and Voting Rule Selection Sensitivity Measures of Coherence and Voting Rule Selection Sensitivity Weak Measures and WSR Selection Sensitivity Strong Measures and WSR Selection Sensitivity Other Voting Rules Approval Voting Lottery Based Voting Rules Median Voting Rule Conclusion Complete PMR Ranking Efficiencies Introduction Candidate Ranking Sensitivity to WSR Selection Empirical Results Probability Representations for the Same WSR Ranking Condorcet Ranking Efficiency Empirical Results The Impact of Social Homogeneity The Presence of a PMR Cycle The Impact of Group Mutual Coherence Condorcet Committees Committee Election Paradoxes Condorcet Committee Definitions Condorcet Committee Efficiency Condorcet Committee Efficiency Summary 366 References 367 Index 381