With most colleges requiring students to take at least one mathematics class, faculty members have a responsibility to design courses that are both sufficiently non-trivial to be considered a college course and accessible to non-majors who may hate or fear mathematics. When, two years ago, I was asked to revamp the syllabus of Math 105, one of the “quantitative literacy” courses at Iowa State University, I quickly discovered a number of recent books in which mathematical ideas are applied to problems in politics or social choice. The idea of doing this is not recent, and goes back at least 25 years to the publication of the first edition of *For All Practical Purposes, *written by COMAP (the Consortium for Mathematics and its Applications), which just recently came out in a 9^{th} edition. Another book along the same lines, which also devotes several chapters to these ideas, is Peter Tannenbaum’s *Excursions in Modern Mathematics*, which I used the first semester I taught Math 105, but which, unfortunately, did not contain quite enough material in the four chapters devoted to this topic to cover an entire semester’s work. I now use *A Mathematical Look at Politics* by Robinson and Ullman, which seems to be working quite well, and also have looked at other books on this topic such as *Mathematics and Politics* by Taylor and Pacelli, *The Mathematics of Voting and Elections: A Hands-On Approach* by Hodge and Klima, and *Mathematics of Social Choice* by Borgers. As all of these books demonstrate, mathematical reasoning can be used to shed light on interesting problems in politics, specifically including the strange things that can happen in the area of voting. It is precisely because of my involvement in this course that I was eager to look at the book under review; the title had me hooked from the beginning.

Belenky’s book, which focuses on one particular aspect of voting theory (the way in which this country elects a President) is not really a book like any of the others listed above, and is also not the usual kind of book reviewed in this column. For one thing, I doubt this book is intended to be a text; there are no exercises, for example, and the level of exposition is sufficiently high that it reads more like a monograph than a textbook.

The first forty percent or so of the book, though giving some mathematical examples, seems to be primarily addressed to political scientists or students of the Constitution; there is a great deal of history and Constitutional analysis presented here (including discussion of relevant Supreme Court precedent) that is not inherently mathematical. The American presidential voting system (which is unique among countries) presents a lot of interesting issues for discussion, and the author, who has written several other books on the subject, does a good job of discussing them, though this book (in common with many math textbooks that *are* routinely reviewed in this column) cannot be read like a novel. Belenky has definite opinions and is not shy about expressing them, but he provides well-reasoned analysis to support these opinions rather than appeals to emotion or vague feelings about “what ought to be the case”.

To give a sense of what some of these issues are, let me begin with a crash course in the American electoral college system. Most well-educated adults (but not, sadly, quite a few of my students) are at least vaguely aware of the electoral college system and the fact that (as exemplified in the Bush/Gore election of 2000, and three times before that) a candidate can lose the Presidency despite winning a plurality or even majority of the direct popular vote. A number of people feel that this fact is reason enough to jettison the electoral college system, but the arguments made in support of this proposition often fail to take into account the history of the electoral college system and the deliberate compromises made by the Framers of the Constitution in formulating this system. In fact, however, the odd results that can possibly exist go beyond the fact that the popular vote winner can lose the election; it is here where we need to look at the precise mechanics of the system, which are often not fully understood even by people who are generally familiar with the idea of the electoral college.

For example, I suspect quite a lot of people believe that any Presidential candidate who wins a plurality of the popular vote in a state is automatically entitled to be awarded the electoral votes allocated to that state. In fact, while this “winner take all” approach (discussed in chapter five of the book under review) is currently the system generally in effect, it is not Constitutionally mandated.

Every state appoints electors, who cast ballots for the President; these are the electoral votes. Prior to the Twelfth Amendment to the Constitution, an elector submitted two independent ballots, a system which could result in as many as three people each winning a majority of the votes cast. (The Twelfth Amendment changed this by requiring each elector to cast one vote for President and another vote for Vice-President.) The question of whether these electors are obliged to cast their ballots in accordance with the wishes of a majority of the voters is not an easy one to answer; respectable arguments, involving analysis of applicable, but not dispositive, U.S. Supreme Court precedent can be made for both sides. In fact, there is even historical precedent (as recently as the Gore/Bush election) for electors to abstain and cast a blank ballot, a practice which Belenky believes to be probably legitimate under the Twelfth Amendment of the Constitution. The problem of “faithless electors” is but one of a number of “weird outcomes” that Belenky catalogs in chapter four of the book (“What if? Constitutional puzzles, weird outcomes and possible stalemates in Presidential elections.”).

Dissatisfaction with the fact that a popular vote winner may well be an electoral vote loser has resulted in widespread attempts to change the electoral vote system. Direct attempts to amend the Constitution have never been successful, but a proposal that has garnered a lot of attention is the “National Popular Vote” movement, which, if adopted, would result in a set of states that control (between them) more than half of the total number of electoral votes entering into a “compact” and agreeing to cast their votes in accordance with the nationwide popular vote outcome, rather than the outcome in any one state. Belenky is opposed to this plan and discusses its perceived shortcomings in chapter six of the book. (The question of whether this proposal is constitutional would make for a fascinating law school final exam hypothetical.)

After eight chapters (comprising, as noted earlier, about forty percent of the book), Belenky provides ten appendices of rather dense, often research-level, mathematics, addressing such questions as the use of integer programming to solve the question of what minimum percent of the popular vote is necessary to elect a President (Appendix 1), the use of operations research in campaign strategy (Appendix 5), and game-theoretic models for Presidential campaigns (Appendix 9). (Some of this material is addressed in more accessible terms in chapter 8, which is devoted to the question of how election rules can dictate campaign strategy.) The ten appendices require considerable attention and thought even by professional mathematicians, and are almost surely completely beyond the ability of people without mathematical training; this is, of course, by design: it is the first eight chapters of the book that Belenky intends to be read by laypeople.

In view of my expressed motivation for asking to review this book, a natural question on which to end this review is: how useful will this book be when I next teach Math 105 in the upcoming fall semester? The answer is: not very, but that’s not a criticism of the author, since he clearly did not intend it to be used in that manner. However, the first eight chapters did provide me with some interesting reading and food for thought, and when I next discuss the electoral college in my class I will now have more to say to the students than just giving a simple example of how a candidate can win the popular vote and lose the Presidency.

Mark Hunacek (mhunacek@iastate.edu) currently teaches mathematics at Iowa State University, but also spent most of his adult life practicing law, though without getting involved in election law issues.