Processes of collective decision making are seen throughout modern society. How does a government decide on an investment strategy within the health care and educational sectors? Should a government or a community introduce measures to combat climate change and CO2 emissions, even if others choose not too? Should a country develop a nuclear capability despite the risk that other countries may follow their lead?
This introductory text explores the theory of social choice. Social choice theory provides an analysis of collective decision making. The main aim of the book is to introduce students to the various methods of aggregating the preferences of all members of a given society into some social or collective preference. Written as a primer suitable for advanced undergraduates and graduates, this text will act as an important starting point for students grappling with the complexities of social choice theory. With all new chapter exercises this rigorous yet accessible primer avoids the use of technical language and provides an up-to-date discussion of this rapidly developing field.


PREFACE TO THE REVISED EDITION
In her otherwise very positive review of the first edition of this Primer in the
journal Social Choice andWelfare (Vol. 30, 2008),Antoinette Baujard deplored
the absence of exercises. This criticism was more than justified, since a primer
is written for beginners. It is especially important that they find out for themselves
whether they have correctly understood the concepts introduced in the
text, and furthermore some of the logical inferences may be uncommon to
them. In this revised edition, the reader will find 8–10 exercises at the end of
each of Chapters 1–9. Some of the exercises are very easy, and are just intended
to make sure that the reader has thoroughly understood what is discussed in
the text, others are a bit ‘trickier’. Work with pencil and paper can be very
illuminating and rewarding. Hints toward a solution of some of the exercises
are gathered at the end of the book. I am grateful to Nick Baigent and John
Weymark for allowing me to take a look at some of the exercises that they
devised for their own course in social choice theory.
Otherwise, this new edition sees a few minor additions and amendments that
are meant to lead to greater clarity. I wish to thank both teachers and students
of collective choice theory for the comments and suggestions that they made
on the first edition. These were very encouraging for me. In particular, I wish
to thank Greg Fried for his observations in relation to the Arrow–Sen proof
discussed in section 2.2. Finally, I am much indebted to OUP, especially to
Sarah Caro, for making this new edition possible.
Osnabrьck and London
March 2009 Wulf Gaertner

CONTENTS
PREFACE TO THE REVISED EDITION vii
PREFACE AND ACKNOWLEDGEMENTS TO THE FIRST EDITION viii
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 14
1.7 Some exercises 16
2 Arrow’s impossibility result 19
2.1 The axiom system and the theorem 19
2.2 The original proof 21
2.3 A second proof 25
2.4 A third diagrammatic proof 28
2.5 A short summary 34
2.6 Some exercises 34
3 Majority decision under restricted domains 37
3.1 The simple majority rule 37
3.2 Single-peaked preferences 43
3.3 Other domain conditions: qualitative and quantitative 49
3.4 A short summary 53
3.5 Some exercises 53
4 Individual rights 57
4.1 Sen’s impossibility of a Paretian liberal 57
4.2 Gibbard’s theory of alienable rights 59
4.3 Conditional and unconditional preferences 62
4.4 Conditional and unconditional preferences again: matching pennies
and the prisoners’ dilemma 64
4.5 The game form approach to rights 66
x CONTENTS
4.6 A short summary 70
4.7 Some exercises 71
5 Manipulability 75
5.1 The underlying problem 75
5.2 The Gibbard–Satterthwaite theorem 80
5.3 Strategy-proofness and restricted domains 85
5.4 A short summary 92
5.5 Some exercises 93
6 Escaping impossibilities: social choice rules 97
6.1 The Pareto-extension rule and veto power 97
6.2 Scoring functions and the Borda rule 102
6.3 Other social choice rules 109
6.4 A parliamentary vote: Berlin vs. Bonn 114
6.5 A short summary 117
6.6 Some exercises 117
7 Distributive justice: Rawlsian and utilitarian rules 121
7.1 The philosophical background 121
7.2 The informational structure 122
7.3 Axioms and characterizations 124
7.4 Diagrammatic proofs again 129
7.5 Harsanyi’s utilitarianism 134
7.6 A short summary 136
7.7 Some exercises 137
8 Cooperative bargaining 139
8.1 The bargaining problem 139
8.2 Nash’s bargaining solution 140
8.3 Zeuthen’s principle of alternating concessions 148
8.4 The Kalai–Smorodinsky bargaining solution 151
8.5 A philosopher’s view 155
8.6 Kalai’s egalitarian solution 157
8.7 A short summary 159
8.8 Some exercises 160
CONTENTS xi
9 Empirical social choice 163
9.1 Theory and opinions of the general public 163
9.2 Needs vs. tastes – the approach by Yaari and Bar-Hillel 164
9.3 Rawls’s equity axiom – how does it fare? 171
9.4 From here to where? 177
9.5 A short summary 179
9.6 Some exercises 179
10 A few steps beyond 181
10.1 Social choice rules in continuous space 181
10.2 The uniform rule 187
10.3 Freedom of choice 192
10.4 An epilogue instead of a summary 199
REFERENCES 201
HINTS TO THE EXERCISES 209
AUTHOR INDEX 213
SUBJECT INDEX 215