Chaotic Elections! A Mathematician Looks at Voting [Paperback]

Donald G. Saari (Author)

Book Description

ISBN-10: 0821828479 | ISBN-13: 978-0821828472 | Publication Date: April 3, 2001

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.

Editorial Reviews

Review

"...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: 9.7 x 6.9 x 0.4 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: #687,635 in Books (See Top 100 in Books)

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Customer Reviews

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15 of 16 people found the following review helpful:

4.0 out of 5 stars Great on math, weak on policy, March 13, 2002

By 

Alexander R. Small (Pomona, CA United States) - See all my reviews
(REAL NAME)   

This review is from: Chaotic Elections! A Mathematician Looks at Voting (Paperback)

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. If everybody is honest his analysis gives excellent advice on running elections. However, his methods have loopholes that voters and candidates can exploit, trying to get the best outcome even if it means indicating an insincere preference order.

Ultimately, the study of elections has to balance two different approaches: Understanding what the people want based on the info they give, and understanding whether the system gives them incentives to give insincere info. Saari has great insight into the first part, but he doesn't seem as interested in the second part.

11 of 11 people found the following review helpful:

5.0 out of 5 stars An Insightful Look at Voting, March 15, 2002

By 

mrliteral - See all my reviews
(TOP 1000 REVIEWER)    (VINE VOICE)   

This review is from: Chaotic Elections! A Mathematician Looks at Voting (Paperback)

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.

11 of 11 people found the following review helpful:

5.0 out of 5 stars Election Math made Accessible: Cures Headaches, too!, November 28, 2001

By 

Steve Barney - See all my reviews

This review is from: Chaotic Elections! A Mathematician Looks at Voting (Paperback)

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. I guarantee that you will learn something worthwhile from it.

Let me add one last personal note for Wisconsin readers. I am briefly cited in this book in regard to the use of an alternative to the plurality voting procedure, called the 'Second Choice' procedure, in state primaries of Wisconsin's 'Progressive Era.' The use of such alternatives in the history of Wisconsin, and other states, proves that such electoral reforms are actually possible, given a sufficiently educated and motivated electorate. Saari has done the analysis, and made it accessible. Now, it is up to us to educate ourselves, and other voters, and then, in the light of our newfound knowledge, to demand more democratic election procedures.

Also recommended: Principles of Electoral Reform