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The measurement of voting power: theory and practice, problems and paradoxes. (English) Zbl 0954.91019
Cheltenham: Edward Elgar. xviii, 322 p. (1998).
The book under the review is a monograph devoted to the theory of a priory voting power – one of the most significant aspects of voting theory, which, in its own turn, is one of the fundamentals of decision theory. The significance of the investigated problem is justified by the results of K. Arrow (1951) that there doesn’t exist, in principle, a completely democratic formal decision procedure which completely takes into account the interests of each member of a voting group. To help the readers to understand the meaning of the results, the area and the restrictions of their applicability the authors skillfully share a critical analysis of the foundations, examination of methodological presuppositions and a detailed investigation of real voting schemes through numerous cases studies. It is worth noting the way of dealing with mathematical models, accepted by the authors: nothing superfluous and only if this is necessary. This friendly position helps the beginners to understand the subject under research to a high extent.
The book consists of eight chapters: 1. Introduction; 2. Groundwork of the Theory; 3. Power as Influence; 4. Weighted Voting in the US; 5. Weighted Voting in the CMEC; 6. Power as a Prize; 7. Paradoxes and Postulates; 8. Taking Abstention Seriously, and two appendixes: A. Numerical Examples; B. Axiomatic Characterizations.
Chapter 1 consists of a retrospective analysis of the subject under research.
In Chapter 2 the basic mathematical models, namely a Simple Voting Game (SVG) and its dual are determined and examined. Basic compositions of SVG’s are presented.
Chapter 3 deals with the I-power, i.e. a characteristic of a possibility of a member’s vote to influence the outcome of a division. Basic parameters (i.e. indices and measures of voting power, sensitivity, majority deficit etc.) are determined and their properties are investigated. Problems of equalizing the indirect voting powers and of maximizing the sensitivity for two-stage voting processes are solved via exploring the two Square-Root Rules.
Chapter 4 deals with an analysis of the Voting System in the United States. The Principles of elections on the Global (Federal) level (i.e. of the President, the Congress etc.) as well as the Principls of forming of Local Governments (i.e. of Counties, Cities etc.) are analysed. Detailed examination of situations in Nassau County (NY) is presented in the role of a Case Study.
In Chapter 5 the Voting System in CMEC is analysed in detail.
Chapter 6 deals with the P-power, i.e. a measure of estimated or expected share of the fixed purse (in other words, of expected payoff). In terms of cooperative games with transferable utility the Shapley-Shubik index is determined and its basic properties are investigated. The Deegan-Packel and Jonston indices are examined.
In Chapter 7 a detailed examination of a number of paradoxes (of a large size, of redistribution, of new members, quarrelling, of weighted voting, the meet, the transfer, of blocker’s share) is presented.
Chapter 8 consists of the extension of the results presented in Chapters 2, 3 and 6 under supposition that ternary voting rules are admitted.
Appendix A consists of some numerical examples illustrating the technique of calculating of basic indices.
In Appendix B basic properties of characteristic functions with fixed assembly are presented.
The book under review is of exceptional interest for a wide range of potential readers: 1) for students and postgraduates as a basic textbook and a handbook; 2) for lecturers delivering courses connected with Decision Making as the source of mathematical models and real examples; 3) for researchers as a handbook as well as an inexhaustible source of unsolved problems.

91B12 Voting theory
91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance