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Chaotic Elections! A Mathematician Looks at Voting [Paperback]

Donald G. Saari
4.5 out of 5 stars  See all reviews (4 customer reviews)

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Book Description

April 3, 2001 0821828479 978-0821828472
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. <P>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. <P>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. <P>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.

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Editorial Reviews


"...take a look at this interesting book. It will cause you to reexamine ranking and voting methods that you may have come to rely on." ---- Math Horizons

"The book presents a very clear picture of how the author views the central issues of voting theory and provides an excellent entrée into his work." ---- Zentralblatt MATH

"This exceedingly timely and lively book is a mostly non-technical, highly personalized account of author Don Saari's views on and contributions to voting theory and practice. It has perhaps as many surprises and subplots as the extraordinary 2000 United States presidential election ... Some of the surprises are mathematical ones, such as the robustness of the various paradoxes that pervade voting theory. The relevance of chaotic dynamics to these matters is intriguing and gives the book title a delightful double meaning ... written with flair and imagination, making it entertaining and interesting to read ... Saari has written an original, topical, and enjoyable book combining thoughtful social commentary with interesting and accessible mathematics. Read it and read it soon so that you can expand your mathematical horizons, upgrade your civic awareness, and sparkle at social events." ---- MAA Online

Product Details

  • Paperback: 159 pages
  • Publisher: American Mathematical Society (April 3, 2001)
  • Language: English
  • ISBN-10: 0821828479
  • ISBN-13: 978-0821828472
  • Product Dimensions: 7.2 x 10 inches
  • Shipping Weight: 11.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #799,592 in Books (See Top 100 in Books)

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17 of 18 people found the following review helpful
4.0 out of 5 stars Great on math, weak on policy March 13, 2002
First, I'm a physicist, so the math was fine for me, but some people might find it frustrating. However, if you're willing to sink your teeth into it you'll get the important ideas.
Second, Saari's insights into the role of symmetry in three-person elections are beautiful. He shows that even if none of the three (or more) candidates can beat all of the others head-to-head there is still interesting information present. When resolving cyclic ambiguities (George beats Bill, Bill beats Ross, Ross beats George) Saari's mathematical insights may be quite useful.
However, I have to take Saari to task for his criticism of Approval Voting (where you simply indicate yes or no for each candidate). He points out that the Approval winner cannot be predicted based solely on people's preference orders (e.g. I might like Ross better than Bill or George, and Bill better than George). Saari sees this as a defect, because "anybody could win." Approval, however, makes use of different information. Which of those candidates pass your threshold? Vote yes for all that you find acceptable.
Seen in this light, Approval is a perfectly rational policy.
Also, Saari doesn't think too highly of the Condorcet criterion: If one candidate can beat all others in one-on-one contests then that candidate should win. It is true that sometimes no candidate meets that criterion, and in those cases Saari's analysis provides important insights on how to resolve the situation. However, sometimes there is in fact one person who can beat all others one-on-one. In that case, no amount of analysis can change the fact that the Condorcet candidate is preferred over all others, and should win.
Finally, Saari gives short shrift to strategic considerations.
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12 of 12 people found the following review helpful
5.0 out of 5 stars An Insightful Look at Voting March 15, 2002
Saari has put together a small but useful book on the trickiness inherent in voting and the potential paradoxes that can get minor candidates elected. In races with more than two candidates, the method of vote counting is extremely important, and it is possible that by changing the counting process, you could have any candidate you want winning. Theoretically, an election could be rigged without a single dishonest vote.
The obvious solution is to choose the counting method before the election, not afterwards, but there are perils to watch out for nonetheless. Saari goes into depth about these dangers.
This depth often gets very technical, something Saari is up front about. Less than half the book is really aimed for the lay reader; the remainder is aimed more towards mathematicians. Nonetheless, I recommend this book for everyone interested in the democratic process; even if you can't get into the math, there are still enough important insights to make this book more than worthwhile.
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12 of 12 people found the following review helpful
5.0 out of 5 stars Election Math made Accessible: Cures Headaches, too! November 28, 2001
This book makes some of Saari's most important research into the mathematics of voting accessible to a general audience for the first time. Personally, I greatly appreciate this book. With very little math background, I have tried in vain for the past year to comprehend some of Saari's research articles, with the assistance of a math professor. The result was very little comprehension, and a lot of headaches. This book has, for the first time, enabled me to understand some of the things I had struggled with in vain for months, and greatly increased my comprehension of many of those things I saw only dimly before. I thank the author for the mercy he has shown towards me, and similarly interested readers of his work.
At this time, Saari seems to be the world's leading researcher in the mathematics of voting and group decision making. While most of the general public, in the US at least, has remained almost entirely ignorant of the paradoxes of voting, mathematicians have recognized and struggled with them for centuries, since they recognized that the widespread rule that 'a plurality shall elect' can result in the election of the voters' least preferred candidate; for example, when there are 3 candidates, the plurality winner may be a candidate who is the last choice of up to two-thirds of the voters. Saari's recently published research papers, which resolve many of these profoundly difficult mind stumpers, and the recent US Presidential election, not to mention the begging and pleading of mathematical simpletons like myself, combined to motivate Saari to write this book.
If you are at all interested in having your vote properly accounted for in everything from selecting your group's next officer, to future national elections, I recommend this book to you.
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5 of 5 people found the following review helpful
Format:Paperback|Verified Purchase
Saari's book, as he states in the title and his introduction, is very mathematical. This does not bother me; my own background in science gives me strong enough math to handle this book, though more purely social-science-oriented people may find the book too heavy on the mathematics to help them.

The book analyzes a number of voting systems, and makes a good case for his point that the procedure of casting and counting votes will often determine the result more than the actual preferences of the voters. And he comes up with some conclusions with which I agree: the weakness of approval voting (though his reasons for disliking it differ from mine) and the insufficiency of Banzhaf's method of calculating power in a weighted voting system, for example.

But he generally makes one assumption which does not adequately reflect reality. This assumption is that an ordinal sequence of preference is an adequate representation of a voter's wishes (and that, in consequence, how much a voter prefers candidate A to candidate B is measured by how many candidates are between them on a list of preferences). In fact, with enough candidates in a race, there are bound to be several that a voter likes almost equally well, others that a voter dislikes equally well, and probably others about whom the voter is indifferent (possibly because he knows nothing about them!) Thus, there may be a huge gap between a voter's opinion of, for example, his fourth and fifth preferences, while he may like the fourth one almost as much as his top preference. As a result, Saari ends up favoring a system, the Borda count, which gives absolutely equal differences between the values assigned to consecutively numbered preferences, with no recognition (at least until the next-to-last page of the book!
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