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Voting paradoxes and how to deal with them. (English) Zbl 0943.91030

Berlin: Springer. x, 153 p. (1999).
Social choice theory has been developed enormously in the last twenty-five years. As the professional field has expanded, so it has also garnered increasing attention from those outside the field. One of the more beguiling features of social choice theory is the presence of paradoxes or counterintuitive behaviour. In this excellent non-technical text, the author has produced an accessible survey of many of the paradoxes that pervade voting theory. His clear style and numerous simple examples will encourage a wide readership.
The author covers a wide range of paradoxes in social choice theory, from the well-known Condorcet paradox (where a group of voters prefer \(A\) to \(B\), \(B\) to \(C\) and \(C\) to \(A\)), through no-show paradoxes (where a voter achieves a more desirable result by not voting), and aggregation paradoxes such as Simpson’s, to representation paradoxes such as the Alabama or house monotonicity paradox. He also summarizes the research on the paradoxical act of voting itself: why do people vote when an individual vote has so little impact on an election? Each of the many topics is concisely introduced and explained, with brief references to accessible literature, and illustrated with carefully chosen simple examples. The abundance of these examples is a particular strength of the book. In his analysis of the paradoxes, the author clearly indicates which are procedure-specific, the scopes of the paradoxes, and how to avoid them. Where results are known, he summarizes research on the probabilities of certain paradoxes occurring under various assumptions. A nice historical chapter describes the pioneering work of Borda and Condorcet, including problems of interpreting their works.
Throughout, The author is clear and non-technical. In discussing Saari’s geometric approach to voting, he simplifies to the case of only three candidates, using what he terms ‘Saari triangles’ to provide a clear visualization of the results. Similarly, in discussing probabilistic issues, he requires only basic mathematics and eschews measure theory.
The author summarizes well the literature on each topic and indicates the boundaries of current knowledge. His bibliography provides a valuable entrée to the field for a newcomer.

MSC:

91B12 Voting theory
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91B14 Social choice
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