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Chaotic Elections! A Mathematician Looks at…
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Chaotic Elections! A Mathematician Looks at Voting

by Donald G. Saari

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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. ( )
  dougb56586 | Apr 12, 2016 |
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. ( )
  FPdC | May 24, 2010 |
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Amazon.com Product Description (ISBN 0821828479, Paperback)

What does the 2000 U.S. presidential election have in common with selecting a textbook for a calculus course in your department? Was Ralph Nader's influence on the election of George W. Bush greater than the now-famous chads? In Chaotic Elections!, Don Saari analyzes these questions, placing them in the larger context of voting systems in general. His analysis shows that the fundamental problems with the 2000 presidential election are not with the courts, recounts, or defective ballots, but are caused by the very way Americans vote for president.

This expository book shows how mathematics can help to identify and characterize a disturbingly large number of paradoxical situations that result from the choice of a voting procedure. Moreover, rather than being able to dismiss them as anomalies, the likelihood of a dubious election result is surprisingly large. These consequences indicate that election outcomes--whether for president, the site of the next Olympics, the chair of a university department, or a prize winner--can differ from what the voters really wanted. They show that by using an inadequate voting procedure, we can, inadvertently, choose badly. To add to the difficulties, it turns out that the mathematical structures of voting admit several strategic opportunities, which are described.

Finally, mathematics also helps identify positive results: By using mathematical symmetries, we can identify what the phrase "what the voters really want" might mean and obtain a unique voting method that satisfies these conditions.

Saari's book should be required reading for anyone who wants to understand not only what happened in the presidential election of 2000, but also how we can avoid similar problems from appearing anytime any group is making a choice using a voting procedure. Reading this book requires little more than high school mathematics and an interest in how the apparently simple situation of voting can lead to surprising paradoxes.

(retrieved from Amazon Thu, 12 Mar 2015 18:08:23 -0400)

What does the 2000 U.S. presidential election have in common with selecting a textbook for a calculus course in your department? Was Ralph Nader's influence on the election of George W. Bush greater than the now-famous chads? This expository book shows how mathematics can help to identify and characterize a disturbingly large number of paradoxical situations that result from the choice of a voting procedure. Saari's book should be required reading for anyone who wants to understand not only what happened in the presidential election of 2000, but also how we can avoid similar problems from appearing anytime any group is making a choice using a voting procedure.… (more)

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