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Chaotic Elections! A Mathematician Looks at Voting

Donald G. Saari
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Edward W. Packel
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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, which no doubt prompted the AMS to encourage the writing of such an informal and advocative manifesto.

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. But there are also quantitative and qualitative lessons in history and politics as well as a not-so-hidden agenda involving a long running controversy about preferred voting methods (more about this later). While the book states quite a few interesting theorems, many of them by the author, it sensibly and without apology makes no effort to present or even discuss proofs. These can be found by effective references and an eighty item bibliography.

In the spirit of full disclosure I confess that like Saari, a friend and early 1980s co-author, I too was willingly seduced in mid-career by the delights, challenges, and "impossibilities" of voting theory, known more formally as social choice theory. Consequently, I am favorably disposed towards the subject matter and prepared to fill in some of the inevitable mathematical gaps in an ambitious undertaking of this kind. The Preface suggests that general readers should try to digest the first part of each chapter, skipping to the next chapter when the going seems to get and stay rough. This is sound and realistic advice.

Some of the main themes treated in the book are the following:

  • Various voting methods defined, illustrated, and compared
  • Paradoxical aspects of voting (general and specific)
  • The Borda Count versus Approval Voting
  • Application of above ideas to the 2000 election and many others
  • Saari's theoretical results on strategic voting, cycles and symmetry, and "chaos"
  • Comments and suggestions about voting procedures used by different societies (including USA, MAA, and AMS).

The fundamental issue in voting theory is how one might assign to any given voting profile (a vector giving the individual preferences over a set of options for a set of voters) a societal ranking of those options. Since it makes sense to require such an aggregation procedure to be spelled out in advance and to be deterministic, a voting method essentially becomes a function mapping profiles into societal preferences. According to Saari, the most effective such procedure is the Borda Count (BC), which works in the following natural way. If there are n options or candidates, a voter's top choice is given n-1 points, with the points awarded decreasing from there — i.e., the kth ranked choice is given n-k points. The societal ranking is then obtained by adding the points assigned to each candidate by all the individual voters.

The other main protagonist in the book is the Approval Voting method (AV). This method allows each voter to give a point to each "approved" candidate or option and these points are added to get the societal ranking. Mathematical political scientist Steve Brams and his AT&T Labs coauthor Peter Fishburn have been instrumental in analyzing and promoting AV and it has become the method of choice for multicandidate elections in a number of universities and professional societies (including the AMS and the MAA).

At more than a few places in the book, Saari presents his case for the marked superiority of BC over AV and other voting methods. More full disclosure: Steve Brams and I are old friends and this gives me a political reason to withhold my vote in this 15 year-old skirmish (is there a word halfway between debate and feud?). In fact, I have a better reason to abstain — both methods have compelling arguments to support them, mathematical and otherwise. Indeed, this may be yet another manifestation of a common theme in game theory and social choice theory. There are no "best" solution concepts, voting methods, or power indices. Evidence for this comes from the theorems of Arrow and Gibbard-Satterthwaite (both discussed to some extent by Saari) and the ubiquitous paradoxes and anomalies that are present with all proposed methods (dramatically demonstrated in the book). My reaction to all of this, including the BC/AV debate, is one of pride and appreciation. Such indeterminacy adds richness to mathematics and it is refreshing to see genuine controversy in modern day applicable mathematics.

Chaotic Elections! is written with flair and imagination, making it entertaining and interesting to read for anyone with a solid undergraduate mathematics background and an interest in strategic and electoral matters. To get us thinking, Saari recasts social choice in terms of ranking students who have been given grades by various professors (it is no surprise that he equates the standard GPA method with BC and gives this top marks). And just as Senator Jeffords of Vermont was leaving the Republican party last week, I found myself reading Saari's prophetic comments on page 8 about the difficulty of maintaining tight party discipline in Congress as he writes "suppose two Republican Congressmen and two Republican Senators leave their party to form a de facto 'Consensus Party'" and notes that this would force both parties to court their support.

I applaud Saari for trying to take advantage of this teachable moment in American history. While a partial solution to what happened in Florida and elsewhere during the remarkably close 2000 national election may come from modernizing our voting hardware and avoiding butterfly ballots, this book makes a strong case that we should think more deeply about our voting methods and that mathematics can play a major role in guiding this thinking.

While the book was, by necessity, written in a short amount of time, it does not appear to suffer from this. I found very few errors or other structural shortcomings. One point of confusion appears on page 30, where Saari's statement of Arrow's Theorem refers to "the above five conditions," only two of which were evident. And I could not help noticing that a bibliographic reference dropped my first initial, renaming me (presidentially?) as "W" Packel. There is a short index that is helpful but could be more complete. Though some may feel that, for an expository work, the book is rather heavily laden with Saari's own results, I have no problem with this. Indeed, the author's work over the past two decades has revolutionized important aspects of voting theory and has earned him the right to tell his own tale.

In conclusion, 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. And beware — you too might be gripped by social choice fever and find yourself prospecting in this mathematically rich field of study.

Edward W. Packel ( has been teaching mathematics and computer science at Lake Forest College for the past thirty years, with occasional time off for good behavior. His mathematical research interests have included functional analysis, game theory, information-based complexity and the use of technology in teaching mathematics. Every July he teaches the Rocky Mountain Mathematica workshop in Colorado with Stan Wagon. He is the author of The Mathematics of Games and Gambling (MAA, 1981) and several other texts.

  • A mess of an election
  • Voter preferences, or the procedure?
  • Chaotic election outcomes
  • How to be strategic
  • What do the voters want?
  • Other procedures; other assumptions
  • Bibliography
  • Index