##
**Axioms of cooperative decision-making.**
*(English)*
Zbl 0699.90001

Econometric Society Monographs, 15. Cambridge etc.: Cambridge University Press. xiv, 332 p. £40.00; £49.50 (1988).

This is the first book of its kind to the best of the reviewer’s knowledge. It considers a unifying theme, axiomatic cooperative decision making, and applies it to four domains: Welfarism, cooperative games, public decision making and voting and social choice.

About 30 axioms are discussed among which intraprofile axioms such as Pareto optimality and interprofile axioms such as independence axioms (a solution does not change when the parameters change) or monotonicity axioms (a solution shifts in a certain direction related to the shift of the parameters).

Two broad categories of models are studied. The first one, corresponding to Part I-III (welfarism, cooperative games and public decision making), is devoted to sharing divisible surplus models and the second one, Part IV (voting and social choice), to the collective choice of an indivisible public decision.

In Part I, the theory of utilitarianism and modern variants of it are presented. The main assumption is then that only individual utility levels matter when comparisons are made and that these levels are interpersonally comparable. Classical utilitarianism, egalitarianism, inequality measurements and axiomatic bargaining are discussed.

Part II deals with cooperative games. The presentation is highly original since a whole chapter is devoted to the cost-sharing games. Another chapter describes the Shapley value, the nucleolus and core selections.

The public decision mechanisms studied in Part III include cost-sharing models (the author constructs solutions based on the Shapley value and the nucleolus for these models), surplus-sharing models, regulated monopoly (the core being the main concept in this case) and strategyproof mechanisms.

Part IV (voting and social choice) is again very original since the author starts from practical methods (Borda rule, scoring methods, Copeland rule, Simpson rule, plurality with a runoff etc.) and studies their robustness to reasonable (not to say ethical) properties.

The two last chapters are devoted to strategyproofness à la Gibbard- Satterthwaite and core stability and to aggregation of preferences à la Arrow. They include proofs of the two classical impossibility theorems (Gibbard-Satterthwaite’s and Arrow’s) as well as Nakamura’s theorem on the nonemptiness (stability) of the core of voting games (or the acyclicity of the strict preference relation generated by the voting games). The reviewer appreciates this pre-eminence given to Nakamura’s result.

There are two features of the book that are particularly noteworthy. This is already outlined in Amartya Sen’s foreword. First, though, as mentioned before, the classical impossibility results are presented, the author favors throughout positive results, showing hereby that all this literature is applicable. Second, though it is published in a monograph series, and it is a monograph, it can be used as a textbook. The author himself alludes to this fact in the introduction. There is a large number of exercises and, above all, the readibility is exceptional for such difficult subjects. The reviewer is even persuaded that this book is a real pedagogical tour de force. Furthermore, it is beautifully produced thanks to the Econometric Society and Cambridge University Press. It is not only highly recommended but, in the reviewer’s opinion, an obligatory reading for every theoretically-inclined economist or political scientist.

{A slight (bibliographical) mistake is made concerning Nakamura’s theorem. The author refers to a 1975 paper by Nakamura which is about a restricted domain condition introduced by M. Dummett and R. Farquharson [Econometrica 29, 33-43 (1961)]. The correct reference is: K. Nakamura, Int. J. Game Theory 8, 55-61 (1979; Zbl 0415.90087).}

About 30 axioms are discussed among which intraprofile axioms such as Pareto optimality and interprofile axioms such as independence axioms (a solution does not change when the parameters change) or monotonicity axioms (a solution shifts in a certain direction related to the shift of the parameters).

Two broad categories of models are studied. The first one, corresponding to Part I-III (welfarism, cooperative games and public decision making), is devoted to sharing divisible surplus models and the second one, Part IV (voting and social choice), to the collective choice of an indivisible public decision.

In Part I, the theory of utilitarianism and modern variants of it are presented. The main assumption is then that only individual utility levels matter when comparisons are made and that these levels are interpersonally comparable. Classical utilitarianism, egalitarianism, inequality measurements and axiomatic bargaining are discussed.

Part II deals with cooperative games. The presentation is highly original since a whole chapter is devoted to the cost-sharing games. Another chapter describes the Shapley value, the nucleolus and core selections.

The public decision mechanisms studied in Part III include cost-sharing models (the author constructs solutions based on the Shapley value and the nucleolus for these models), surplus-sharing models, regulated monopoly (the core being the main concept in this case) and strategyproof mechanisms.

Part IV (voting and social choice) is again very original since the author starts from practical methods (Borda rule, scoring methods, Copeland rule, Simpson rule, plurality with a runoff etc.) and studies their robustness to reasonable (not to say ethical) properties.

The two last chapters are devoted to strategyproofness à la Gibbard- Satterthwaite and core stability and to aggregation of preferences à la Arrow. They include proofs of the two classical impossibility theorems (Gibbard-Satterthwaite’s and Arrow’s) as well as Nakamura’s theorem on the nonemptiness (stability) of the core of voting games (or the acyclicity of the strict preference relation generated by the voting games). The reviewer appreciates this pre-eminence given to Nakamura’s result.

There are two features of the book that are particularly noteworthy. This is already outlined in Amartya Sen’s foreword. First, though, as mentioned before, the classical impossibility results are presented, the author favors throughout positive results, showing hereby that all this literature is applicable. Second, though it is published in a monograph series, and it is a monograph, it can be used as a textbook. The author himself alludes to this fact in the introduction. There is a large number of exercises and, above all, the readibility is exceptional for such difficult subjects. The reviewer is even persuaded that this book is a real pedagogical tour de force. Furthermore, it is beautifully produced thanks to the Econometric Society and Cambridge University Press. It is not only highly recommended but, in the reviewer’s opinion, an obligatory reading for every theoretically-inclined economist or political scientist.

{A slight (bibliographical) mistake is made concerning Nakamura’s theorem. The author refers to a 1975 paper by Nakamura which is about a restricted domain condition introduced by M. Dummett and R. Farquharson [Econometrica 29, 33-43 (1961)]. The correct reference is: K. Nakamura, Int. J. Game Theory 8, 55-61 (1979; Zbl 0415.90087).}

Reviewer: M.Salles

### MSC:

91B14 | Social choice |

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91A12 | Cooperative games |