CONTENTS
ABOUT THE AUTHOR xiii
1 Introduction 1
1.1 Basic questions 1
1.2 Catching a glimpse of the past 3
1.3 Basic formalism 6
1.4 Aggregation of preferences – how can this be done? 9
1.5 The informational aspect 12
1.6 A path through haze or how to read this book 15
2 Arrow’s impossibility result 17
2.1 The axiom system and the theorem 17
2.2 The original proof 19
2.3 A second proof 23
2.4 A third diagrammatic proof 26
2.5 A short summary 32
3 Majority decision under restricted domains 33
3.1 The simple majority rule 33
3.2 Single-peaked preferences 38
3.3 Other domain conditions: qualitative and quantitative 44
3.4 A short summary 48
4 Individual rights 51
4.1 Sen’s impossibility of a Paretian liberal 51
4.2 Gibbard’s theory of alienable rights 53
4.3 Conditional and unconditional preferences 56
4.4 Conditional and unconditional preferences again: matching pennies
and the prisoners’ dilemma 58
4.5 The game form approach to rights 60
4.6 A short summary 64
5 Manipulability 67
5.1 The underlying problem 67
5.2 The Gibbard–Satterthwaite theorem 72
x CONTENTS
5.3 Strategy-proofness and restricted domains 77
5.4 A short summary 84
6 Escaping impossibilities: social choice rules 87
6.1 The Pareto-extension rule and veto power 87
6.2 Scoring functions and the Borda rule 92
6.3 Other social choice rules 99
6.4 A parliamentary vote: Berlin vs. Bonn 104
6.5 A short summary 107
7 Distributive justice: Rawlsian and utilitarian rules 109
7.1 The philosophical background 109
7.2 The informational structure 110
7.3 Axioms and characterizations 112
7.4 Diagrammatic proofs again 117
7.5 Harsanyi’s utilitarianism 122
7.6 A short summary 124
8 Cooperative bargaining 127
8.1 The bargaining problem 127
8.2 Nash’s bargaining solution 128
8.3 Zeuthen’s principle of alternating concessions 135
8.4 The Kalai–Smorodinsky bargaining solution 138
8.5 A philosopher’s view 142
8.6 Kalai’s egalitarian solution 145
8.7 A short summary 147
9 Empirical social choice 149
9.1 Theory and opinions of the general public 149
9.2 Needs vs. tastes – the approach by Yaari and Bar-Hillel 150
9.3 Rawls’s equity axiom – how does it fare? 157
9.4 From here to where? 163
9.5 A short summary 165
10 A few steps beyond 167
10.1 Social choice rules in continuous space 167
10.2 The uniform rule 173
CONTENTS xi
10.3 Freedom of choice 178
10.4 An epilogue instead of a summary 185
REFERENCES 187
AUTHOR INDEX 195
SUBJECT INDEX 197