Donald G. Saari

In this short book (152 pages), a Mathematics Professor explains how elections can often yield drastically different outcomes depending on which voting procedure is used, and further, that if a low ranked candidate drops out, the ordering of all higher ranked candidates will potentially change. He shows that these problems exist in all procedures that are in common use: e.g. Plurality, Anti-plurality, Approval, Cumulative, the Borda Count, and the Condorcet Count. Numerous simple examples are used to illustrate how easily the election outcomes may change. In addition, he has devised a very nice graphical method which helps visualize the election profiles for the cases of 3 or 4 candidates (it won't work for greater than 4 candidates for the same reason that 4-dimensional space cannot be visualized in 3-dimensions). In addition to these examples, the author also reviews some actual elections: e.g. the Clinton-Bush-Perot presidential election of 1992, the Lincoln-Douglas-Bell-Breckinridge presidential election of 1860, and the voting for the 2000 Olympic Committee City, among others. He also explains “Arrow's Theorem” discovered by Robert Arrow in the 1950s. Since every voting procedure in use was known to sometimes yield unexpected outcomes, Arrow posed the question: “Does a fair voting procedure exist?”, and then shows the answer to be: “no”.

This book is easily read by the general reader, but also contains a fair amount of math. Although probably not as rigorous as a textbook might be, there are 13 theorems (all citing a more rigorous book or journal article for more detail). Due to either lack of educational background or lack of sufficient motivation, I was unable to follow some of the denser mathematical sections. However, the book is well written, so it is usually possible to understand the general idea, even when the math is somewhat glossed over. The only section where this was a problem for me was when the author claimed that the Borda Count may be an exception to Arrow's Theorem, if certain conditions are slightly relaxed. This is interesting and significant feature of the Borda Count. I trust that it is true, but would need to spend a lot more time carefully not glossing over the mathematics in order to understand the effect of the modified conditions.

My only criticism of this book is the organization is sometimes hard to follow, and chapter titles don't help very much. Topics are mentioned early, a question may be posed, and then the reader is told that it will be considered in detail later. However, I still recommended the book to anyone interested in this subject.… (mere)

Donald Saari is a renowned mathematician, former Editor-in-chief of the Bulletin of the American Mathematical Society, and author of important work in two paradoxically distinct subject areas: dynamical systems (mainly n-body problems in Newtonian mechanics) and the mathematics of voting systems, where he is arguably the current foremost world authority. He is also a prolific popularizer of this last topic, having to his credit (up to now) the authorship of three books with varying degrees of mathematical pre-requisites (a fourth is announced for the fall of 2008). The book under review is the least mathematically demanding of them. It is an excelent place to start learning about voting and decision-making procedures and the host of unexpected outcomes, some really aparently paradoxical, that can occur. The book explains in very clear and simple terms the hypothesis and contet of Arrow's and Sen's celebrated theorems, then, along three chapters, it exemplifies and explores what is the reason underlying Arrow's, Sen's, and maby other similar results: the inability of much voting and choice procedures to use the connecting information between the parts and the whose, and the concomitant inability to distinguish between rational and irrational voters. Finally, Saari shows a resolution out of the problem in Arrow's theorem by introducing the notion of intensity of pairwise ranking between alternatives, with which Saari proved (elsewhere) that the Borda Count is a nondictatorial procedure satisfying the analogous Arrow's type conditions. This is an extremely interesting book, with close to nil formal mathematics, but that should be read by everyone interested in the subject (be him a mathematician or otherwise) for its clarity of exposition and the capacity of Saari to explain fine points and difficult problems and results in a transparent way and with a minimal amount of technical requirements.… (mere)

Of the two expository books by Saari on the mathematics of voting systems, this is clearly the more mathematically oriented , although it is not exactly a mathematical text, containing no proofs of the stated theorems but only ilustrative exemples and very clear explanations. Saari also refers the reader to the most relevant contributions in the technical literature, including his very many papers and his excelent mathematical monograph Basic Geometry of Voting. Being a kind of middle of the road text between the Social Sciences and the Mathematical communities, this book can leave some people unsatisfied, not exploring in depth neither of the fields. For me, I found it very interesting and a useful stepping stone between Saari's book Decisions and Elections and his mathematical papers and monograph that every serious student of the field must sooner or later plunge into.… (mere)