##
**Social choice theory. An introduction.**
*(English)*
Zbl 0689.90002

Berlin etc.: Springer-Verlag. 163 p. DM 49.00 (1988).

Generally, an introductory textbook follows more or less the organization of former ones or diverges radically from it. In this case, each alternative was impossible to follow since there is no precursor. The nearest candidates are probably “Welfare economics and social choice theory” by A. Feldman (Martinus Nijhoff, 1980) and “Politics, economics and welfare” by J. Bonner (Wheatsheaf, 1986) but they do not cover the same topics and, at least for the second one, the mathematical level is a lower one. This is not to say that the mathematics in Kelly’s book are demanding. In the author’s words “The emphasis is on the level of maturity; relatively little specific detail from algebra will be used”.

It is rather uncommon that the main contributors to a subject write introductory textbooks. However, in all cases I know, they have written among the best textbooks. I cannot tell that this one is the best because it is the only one (though mathematically speaking it is necessarily the best one). But what I can tell is that it is truly a remarkable piece of work and not only I highly recommend it but I think it should be a compulsory reading for every student following courses on advanced microeconomics, public economics, welfare economics, or even on positive political science.

I taught myself a course on social choice theory a few years ago (before the publication of this book) and I only marginally diverged from Kelly on the choice of topics. A textbook must include impossibility theorems à la Arrow, à la Sen and à la Gibbard-Pattanaik-Satterthwaite, and possibility theorems à la Black on majority rule. They are all here.

The author has chosen to start with four chapters on majority rule. The first chapter is devoted to a discussion of majority rule over two alternatives and include a proof of May’s characterization theorem. Then, majority rule is shown to generate the Condorcet paradox and the problem of the frequency of the paradox is described. Black’s single-peakedness is presented in Chapter 3 as a way to avoid this paradox. However, if the space of alternatives is multidimensional, even with strong assumptions on the individuals’ preferences, majority rule may lead to chaos. Chapter 4 is devoted to chaos through examples. Extensions of majority rule are considered in the next chapter, mainly Copeland rule and plurality rule.

The more abstract analysis starts with Chapter 6 where social choice rules are defined and conditions such as domains conditions, positive responsiveness, Pareto conditions, independence of path are presented and described in detail with numerous examples. A distinctive feature of the book is that the values of social choice rules are choice functions rather than “social preferences”. The five following chapters are mainly devoted to the impossibility theorems mentioned formerly. The discussion of rights in the context of Sen’s theorem on the impossibility of a Paretian liberal and of counterthreats in the context of Gibbard- Pattanaik-Satterthwaite strategy proofness analysis are particularly developed. Finally Chapter 12 treats approval voting as a response to the impossibility of a strategy-proof social choice function and Chapter 13 is devoted to the sensitivity of collective rules to mistakes through concepts like the continuity of rules or the distance between orderings. A mathematical appendix describes sets (naive theory), operations on sets, and methods of proofs.

I have not yet mentioned a spectacular feature of the book. This book includes more than 250 exercises. They are according to the author “... the heart of the book. Social choice criteria are often quite slippery and subtle; experience shows that you are unlikely to understand them unless you get your hands dirty working on the exercises”. The reviewer shares this opinion with the author.

There are a number of misprints and small mistakes (for instance Mitterrand - with two r’s - is not a lady whose first name is Francoise but a gentleman called François; Frege was not Italian but German; an arrow on page 36 should be reversed) all of which can easily be corrected in the second edition.

It is rather uncommon that the main contributors to a subject write introductory textbooks. However, in all cases I know, they have written among the best textbooks. I cannot tell that this one is the best because it is the only one (though mathematically speaking it is necessarily the best one). But what I can tell is that it is truly a remarkable piece of work and not only I highly recommend it but I think it should be a compulsory reading for every student following courses on advanced microeconomics, public economics, welfare economics, or even on positive political science.

I taught myself a course on social choice theory a few years ago (before the publication of this book) and I only marginally diverged from Kelly on the choice of topics. A textbook must include impossibility theorems à la Arrow, à la Sen and à la Gibbard-Pattanaik-Satterthwaite, and possibility theorems à la Black on majority rule. They are all here.

The author has chosen to start with four chapters on majority rule. The first chapter is devoted to a discussion of majority rule over two alternatives and include a proof of May’s characterization theorem. Then, majority rule is shown to generate the Condorcet paradox and the problem of the frequency of the paradox is described. Black’s single-peakedness is presented in Chapter 3 as a way to avoid this paradox. However, if the space of alternatives is multidimensional, even with strong assumptions on the individuals’ preferences, majority rule may lead to chaos. Chapter 4 is devoted to chaos through examples. Extensions of majority rule are considered in the next chapter, mainly Copeland rule and plurality rule.

The more abstract analysis starts with Chapter 6 where social choice rules are defined and conditions such as domains conditions, positive responsiveness, Pareto conditions, independence of path are presented and described in detail with numerous examples. A distinctive feature of the book is that the values of social choice rules are choice functions rather than “social preferences”. The five following chapters are mainly devoted to the impossibility theorems mentioned formerly. The discussion of rights in the context of Sen’s theorem on the impossibility of a Paretian liberal and of counterthreats in the context of Gibbard- Pattanaik-Satterthwaite strategy proofness analysis are particularly developed. Finally Chapter 12 treats approval voting as a response to the impossibility of a strategy-proof social choice function and Chapter 13 is devoted to the sensitivity of collective rules to mistakes through concepts like the continuity of rules or the distance between orderings. A mathematical appendix describes sets (naive theory), operations on sets, and methods of proofs.

I have not yet mentioned a spectacular feature of the book. This book includes more than 250 exercises. They are according to the author “... the heart of the book. Social choice criteria are often quite slippery and subtle; experience shows that you are unlikely to understand them unless you get your hands dirty working on the exercises”. The reviewer shares this opinion with the author.

There are a number of misprints and small mistakes (for instance Mitterrand - with two r’s - is not a lady whose first name is Francoise but a gentleman called François; Frege was not Italian but German; an arrow on page 36 should be reversed) all of which can easily be corrected in the second edition.

Reviewer: M.Salles

### MSC:

91B14 | Social choice |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |