Collective Preference and Choice Book

List of figures xii

Preface xiii

Acknowledgments xvi

Part I Introduction 1

1 The reason for the problems 3

1.1 Exercises 6

1.2 Summary 7

2 Brief overview of the problems 9

2.1 Exercises 15

2.2 Summary 16

3 The relationship between preferences and choice 19

3.1 Preference-driven choice 20

3.2 Rationalizable choice 23

3.3 Exercises 26

3.4 Summary 30

Part II Different Preferences 33

4 Do social preferences exist? Arrow's and Sen's impossibility theorems 35

4.1 The social choice model 36

4.2 Arrow impossibility theorem 39

4.3 The Paretian liberal paradox 50

4.4 Exercises 51

4.5 Summary 58

5 The desirable decision rule: axiomatization 60

5.1 Dichotomous choice and the simple majority rule 60

5.2 The Borda rule 64

5.3 Exercises 68

5.4 Summary 74

6 Rule selection based on compromise with the unanimity criterion 76

6.1 The unanimity criterion 76

6.2 Distance between preferences 78

6.3 The metric compromise with the unanimity criterion 80

6.3.1 The example of the Borda rule 83

6.3.2 The example of the plurality rule 84

6.4 Extensions 84

6.5 Exercises 85

6.6 Summary 91

7 Paradoxes of voting 93

7.1 Condorcet's voting paradox 94

7.2 Condorcet's inconsistency 97

7.3 Violation of the Pareto criterion 99

7.4 Violation of consistency 100

7.5 Violation of consistency in contraction and in expansion 102

7.6 Inverted order paradox 103

7.7 The winner-turns-loser paradox 104

7.8 The no-show paradox 105

7.9 Exercises 106

7.10 Summary 114

8 Majority tyranny 117

8.1 Scoring rules 118

8.2 Majority decisiveness 120

8.3 Resolution of the problem of majority tyranny under sincere voting 123

8.4 Resolution of the problem of majority tyranny under insincere voting 124

8.5 Erosion in the majority principle vs. majority tyranny 126

8.6 The golden scoring rule 127

8.7 Appendix 131

8.8 Exercises 132

8.9 Summary 142

9 The problem of inefficient provision of public goods 146

9.1 The collective decision rule and inefficient provision of a public good 147

9.2 Voting 148

9.2.1 The simple majority rule 151

9.2.2 The dictatorial rule 154

9.2.3 The Borda rule 154

9.2.4 The unanimity rule 154

9.3 Voluntary provision of the public good 155

9.4 Appendix 158

9.5 Exercises 158

9.6 Summary 173

10 Do individuals reveal their true preferences? 176

10.1 Non-truthful revelation of preferences 177

10.2 The impossibility theorem of Gibbard-Satterthwaite 180

10.3 Revelation of preferences and the efficient provision of a public good - the dichotomous case 186

10.3.1 The Groves-Clarke revelation mechanism 188

10.4 Exercises 193

10.5 Summary 197

Part III Identical Preferences, Different Decisional Skills 199

11 Which rule is better: the expert rule or the simple majority rule? Decisional errors in dichotomous choice and Condorcet's jury theorem 201

11.1 The expert rule vs. the simple majority rule: the case of three identical decision makers 202

11.2 The expert rule vs. the simple majority rule: the case of three decision makers with unknown decisional skills 203

11.3 Condorcet's jury theorem 205

11.4 Extensions of Condorcet's jury theorem 207

11.4.1 The superiority of the simple majority rule when the decisional skills of the n voters are unknown 207

11.4.2 The superiority of the simple majority rule when the decisional skills of the n voters are different 207

11.5 Is Condorcet's theorem valid when decisional skills are determined endogenously? 208

11.5.1 The extended model: identical endogenous decisional skills 209

11.5.2 Centrally determined decisional skills 209

11.6 Is Condorcet's theorem valid when individual decisions are insincere? 211

11.6.1 Strategic considerations and insincere voting 212

11.6.2 Condorcet's theorem and strategic considerations 213

11.7 Exercises 218

11.8 Summary 224

12 The optimal decision rule under uncertain dichotomous choice 227

12.1 Dichotomous decision making - the model 227

12.2 The optimal decision rule 230

12.2.1 Qualified weighted majority rule 232

12.2.2 Qualified majority rule 233

12.3 The optimal decision rule in the symmetric case 235

12.3.1 Weighted majority rule 235

12.3.2 Simple majority rule 235

12.3.3 The expert rule 235

12.4 Extensions 236

12.4.1 Dependence of decisional skills on the state of nature 236

12.4.2 Dependent decisions 236

12.4.3 The number of decision makers is not fixed 236

12.4.4 Decisional skills are not fixed 237

12.4.5 Partial information on decisional skills 237

12.4.6 Sequential decision 238

12.4.7 More than two alternatives 238

12.5 Exercises 238

12.6 Summary 243

Bibliography 246

Author index 251

Subject index 253