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**Social choice and multicriterion decision-making.**
*(English)*
Zbl 0602.90001

Cambridge, Massachusetts - London: The MIT Press. VII, 127 p. $ 20.25 (1986).

”... This book intends to derive the lessons of social choice theory possible foundations for multicriterion models effective for one type of decision frequently occurring in industry...” (from the authors’ introduction to the book).

This citation gives a concise and exact characterization of the book under review. It is a small volume, succinctly and lucidly written, not addressed to researchers but to ”users”, mainly to managers dealing with the problem of choosing optimal solutions in a multicriterial situation.

The book consists of three parts: In the first part the multicriterial interpretation of the classical Arrow theorem is discussed. Hundreds of contributions were devoted to this theorem in a quantity of monographs and reviews but the possibility of ”multicriterion interpretation” is very seldom suggested and discussed here explicitly. Such an interpretation is only possible in the case when the domain of the aggregate operator is a set (”profile”) of weak order (”criteria”) and its range is a weak order (”aggregate criterium”) as well. In this case the ”condition of unrestricted domain” (axiom 1), the ”condition of independence of irrelevant alternatives” (axiom 2) and the condition of absence of a dictator are incompatible. The first part of the book is fully devoted to the discussion of the corollaries of this result.

Having in mind the absolute necessity of axiom 2 the authors in the second part of the book are considering different ways of imposing constraints on the admissible profiles violating axiom 1. Under consideration are constraints on profiles introduced by Coom’s, Black’s, Romero’s and some other methods. Special attention is paid to Arrow’s and Black’s conditions of single-peakedness. In this part the number of possible profiles which satisfy each of these conditions is calculated, i.e. the estimation of the ”infringement grade of axiom 1” is given.

In part three an array of criteria is examined as an implicit assignment of an n-round tournament matrix. A system of axioms is introduced which must allow the application of certain multistep algorithms leading to variants ranking by means of such a matrix. Different algorithms of such kind are considered (Kohler’s, Arrow-Raynaud’s and other algorithms).

The book is clearly written, richly illustrates by examples and can be recommended as a good textbook for ”users” studying this field.

Reviewer’s remark. The authors’ assumption that in a multicriterial situation the axiom 2 is obligatory and that the issue lies in the possible rejection of axiom 1 seems to me controversial. In practice the rejection of axiom 2 is at least as reasonable as the rejection of axiom 1.

This citation gives a concise and exact characterization of the book under review. It is a small volume, succinctly and lucidly written, not addressed to researchers but to ”users”, mainly to managers dealing with the problem of choosing optimal solutions in a multicriterial situation.

The book consists of three parts: In the first part the multicriterial interpretation of the classical Arrow theorem is discussed. Hundreds of contributions were devoted to this theorem in a quantity of monographs and reviews but the possibility of ”multicriterion interpretation” is very seldom suggested and discussed here explicitly. Such an interpretation is only possible in the case when the domain of the aggregate operator is a set (”profile”) of weak order (”criteria”) and its range is a weak order (”aggregate criterium”) as well. In this case the ”condition of unrestricted domain” (axiom 1), the ”condition of independence of irrelevant alternatives” (axiom 2) and the condition of absence of a dictator are incompatible. The first part of the book is fully devoted to the discussion of the corollaries of this result.

Having in mind the absolute necessity of axiom 2 the authors in the second part of the book are considering different ways of imposing constraints on the admissible profiles violating axiom 1. Under consideration are constraints on profiles introduced by Coom’s, Black’s, Romero’s and some other methods. Special attention is paid to Arrow’s and Black’s conditions of single-peakedness. In this part the number of possible profiles which satisfy each of these conditions is calculated, i.e. the estimation of the ”infringement grade of axiom 1” is given.

In part three an array of criteria is examined as an implicit assignment of an n-round tournament matrix. A system of axioms is introduced which must allow the application of certain multistep algorithms leading to variants ranking by means of such a matrix. Different algorithms of such kind are considered (Kohler’s, Arrow-Raynaud’s and other algorithms).

The book is clearly written, richly illustrates by examples and can be recommended as a good textbook for ”users” studying this field.

Reviewer’s remark. The authors’ assumption that in a multicriterial situation the axiom 2 is obligatory and that the issue lies in the possible rejection of axiom 1 seems to me controversial. In practice the rejection of axiom 2 is at least as reasonable as the rejection of axiom 1.

Reviewer: M.Ajzerman

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91B14 | Social choice |

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

90B50 | Management decision making, including multiple objectives |