You are here

The Mathematics of Voting and Elections: A Hands-On Approach

Jonathan K. Hodge and Richard E. Klima
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical World 22
[Reviewed by
Raymond N. Greenwell
, on

In 1987, the Consortium for Mathematics and Its Applications Project (COMAP) changed forever the liberal arts math course with its book For All Practical Purposes. Many applications of mathematics to nontraditional areas, such as political science, first became widely known through this book. It was soon followed by the similar Excursions in Modern Mathematics, by Peter Tannenbaum. I have used the first four chapters of Tannenbaum as the basis for a course on mathematics and political science, but I found these chapters to be insufficient for an entire semester course. Fortunately, there are entire books written for a liberal arts course on the mathematics of political science, such as Mathematics and Politics, by Alan D. Taylor.

The book by Hodge and Klima is an excellent entry into this field. It is based on a course taught by the authors at Grand Valley State University to students with a wide variety of mathematical backgrounds. Chapter 1 considers the virtues of majority rule in an election with just two candidates. Chapters 2 through 4 review some of the methods for resolving elections with more than two candidates. Chapter 5 goes through a proof of Arrow’s Theorem, which essentially says that a perfectly fair voting method is impossible. Chapter 6 studies weighted voting systems and how one can determine whether a voting system is weighted or not. Chapter 7 discusses measures of power in a weighted voting system. Chapter 8 takes a close look at one very important weighted voting system: the Electoral College. Chapter 9 looks at referendum elections, in which voters can vote on various related referenda, in which their opinion on whether or not one proposition should pass might depend on whether or not another proposition passes. The most interesting result here is that the least preferred outcome by all the voters might be the one that wins! Chapter 10 is on the various methods that have been used or considered for congressional apportionment.

The book has plenty of material for a one-semester course. With more time to devote to each topic than either the COMAP or the Tannenbaum text, the text provides a broader and deeper coverage. The down side is that this occasionally makes the presentation less compelling. For example, devoting the entire first chapter to a mathematical analysis of why majority rule is best (and what this means) in an election with two candidates may seem irrelevant to a student who regards this as obvious. In contrast, the COMAP video The Impossible Dream: Election Theory presents a humorous and disturbing scenario of an election with 5 candidates in which 5 different voting methods produce 5 different winners. It then briefly presents Arrow’s Theorem, all in just a half hour. I have found that this approach convinces students on the first day of class that mathematics is relevant. Hodge and Klima lose some of the excitement by slowly releasing this information over three chapters. It’s also curious that with so much detail, one of the most common methods for resolving an election with more than two candidates, plurality with a runoff, isn’t mentioned.

Hodge and Klima have a friendly and clear style that students will appreciate. This doesn’t mean that it’s all easy reading; the proof of Arrow’s Theorem will challenge most students at this level. The authors have tried to capture the spirit of a Moore method course, one consequence of which is the complete absence of worked-out examples. To compensate, many of the exercises (labeled “Questions”), which are scattered throughout each section, have solutions at the end of the section. These questions are denoted with a star. For example, Chapter 10 has 47 questions, 9 with solutions. This may not be enough for some instructors. For example, Chapter 10 introduces the new-states paradox but provides no exercises allowing students to play around with this idea. In contrast, the corresponding chapter in Tannenbaum has 10 worked-out examples and 59 exercises.

Not all the questions are mathematical. For example, Question 2.30 asks the reader to speculate on the result of the 2000 presidential election if John McCain had run as an independent candidate. Question 8.35 asks the reader to summarize the 12th Amendment to the Constitution and to investigate the historical events behind it. Question 8.17 would be a significant project to answer; it asks the student to fill in the blank and justify that number for the following statement: “In an actual U.S. presidential election with only two candidates, it would be virtually impossible for a candidate to win the election without receiving at least ___% of the popular vote.” A satisfactory answer to Question 9.34 might be worthy of publication; it asks for a new method for solving something called the separability problem in a referendum election, an area in which the research is recent and sparse.

Despite this book’s shortcomings, it is still a fine book. It is well-written and well-edited, with virtually no errors. (The errata page on the book’s web site lists two minor corrections.) Every instructor teaching this topic should consider this as the textbook, and should have this book regardless of what textbook is chosen.


For All Practical Purposes: Mathematical Literacy in Today’s World (6th ed.), W. H. Freeman, 2003, ISBN 0-7167-4783-9.

For All Practical Purposes: Social choice: The Impossible Dream: Election Theory , Annenberg/CPB Project, 1986.

Alan D. Taylor, Mathematics and Politics: Strategy, Voting, Power and Proof , Springer-Verlag, 1995, ISBN 0-387-94391-9.

Peter Tannenbaum, Excursions in Modern Mathematics (5th ed.), Prentice Hall, 2004, ISBN 0-13-100191-4.

Raymond N. Greenwell ( is a Professor of Mathematics at Hofstra University in Hempstead, New York. His research interests include applied mathematics and statistics, and he is coauthor of the texts Finite Mathematics and Calculus with Applications, published by Addison Wesley.

* What's so good about majority rule?
* Perot, Nader, and other inconveniences
* Back into the ring
* Trouble in democracy
* Explaining the impossible
* One person, one vote?
* Calculating corruption
* The ultimate college experience
* Trouble in direct democracy
* Proportional (mis)representation
* Bibliography
* Index