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 rale 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 rale ............................. 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
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