It is the height of the presidential primary season as I write this review, with Super Tuesday looming on the horizon. This means that one can barely turn on the television, flip through the radio stations, or fire up a web browser without being inundated with news about the horserace of the elections — the strategies, the game theory, and occasionally even the issues.

Even the mathematical community seems to be in on the fun this year, with "Math And Voting" being the theme of this year's Mathematics Awareness Month . If, like me, the kind of person for whom the ubiquitousness of election talk and strategizing is more exciting than exhausting, then you probably know the name of Steven Brams. Brams is Professor of Politics at New York University, but his interests are very mathematical in nature and he has written many books and articles applying game theory and other forms of quantitative analysis to problems such as fair-division, treaty negotiation, and — yes — presidential elections. His book *The Presidential Election Game* was originally released by Yale University Press in 1978 and was recently re-issued in a nice paperback edition by AK Peters.

Each chapter of *The Presidential Election Game* takes an aspect of presidential elections and constructs a mathematical model which one can use to explore the topic at hand. All of the models discussed are relatively elementary from both a mathematical and political point of view, and while one could quibble with many of the overly simplistic assumptions which go into the model, it is exactly this lack of subtlety and sophistication which makes the book extremely accessible to a wide audience. Reading this book, one could easily imagine it being used in courses in mathematics departments as well as in political science departments, as well as being a book one could pick up for either leisure reading or as an introduction to more serious pursuits.

The first chapter presents a spatial analysis of elections, in which the base assumption is that voters will choose to vote for the candidate who they are closest to on a left-right spectrum. Depending on the distribution of voter preferences as well as the political beliefs of their candidates, the model then assumes that any given candidate will choose their own beliefs in such a way as to maximize the number of votes they will get, and this allows one to ask questions about equilibrium strategies as well as whether it even makes sense to enter the race in the first place. This model may be too cynical for some readers, but it certainly forms a good starting point for any mathematical model of elections. Brams does not go into the more sophisticated analysis which would be required if one wanted to consider more than one axis along which candidates would vary, but he does suggest how one could approach such an analysis.

The next chapter analyzes the building of coalitions of delegates at a nominating convention, and in particular asks the question of when it makes sense for delegates to shift from one candidate to another — questions which feel particularly relevant in this election year where (as of this writing, if not as of the time you are reading this review) the possibility of brokered conventions looms for both Republicans and Democrats. The third chapter looks at several models related to strategizing surrounding the electoral college, and how a candidate might be able to exploit the deviations from a true `one person, one vote' model of democracy that the electoral college provides, as well as the question of how to allocate resources between different states. The models developed in the fourth chapter analyze what the author calls `coalition politics', and the impact that political parties have on the process of elections. The fifth chapter considers the un-election of presidents, and in particular sets up a game theoretic model which the author uses to investigate the conflicts between Richard Nixon and the Supreme Court in the days between the Watergate break-in in Nixon's resignation.

Brams opens the final chapter of the book by commenting that "it is traditional to conclude a book on presidental elections with calls for reform. I shall not depart from this tradition." He then goes on to discuss different methods of voting and in particular how voters might behave differently if they were allowed to cast negative votes for their least favorite candidates in addition to (or instead of) positive votes for their top choices. He then introduces 'approval voting', a system in which each voter can decide whether each candidate is acceptable or unacceptable and the candidate who is acceptable to the most voters wins. The chapter closes with several theorems describing situations in which approval voting is an ideal system — as well as situations in which it is not.

When the original release of *The Presidential Election Game* was reviewed in the American Mathematical Monthly nearly thirty years ago, the reviewer wrote about its usefulness in a course on mathematical modeling: “It can illustrate how even mathematically simple models can lead to useful conclusions, and it should stir up many ideas in the mind of the curious reader.” That review certainly still holds true today — Brams has written an extremely readable book which introduces models that are complicated enough to have interesting things to say while not being so complicated that a student couldn't pick the book up and get something out of it. An extensive bibliography gives references for readers who want to see more complicated models.

In fact, my only significant complaint about the book is the fact that it feels quite out of date. The author wrote a new preface which briefly discusses how the models he developed in the 1970s would deal with the developments of the last several decades, but the bulk of the text has not been updated at all from the original edition. In fact, the very first paragraph of chapter one mentions that “the campaigns of most presidential candidates do not attract wide news coverage, however, until the first caucuses and primaries, which begin in January and February of an election year,” which feels less accurate given that we have been watching nationally televised debates with Obama and Clinton since April 26, 2007 (and the Republicans started the following week).

Most of the changes in the political landscape may not affect the accuracy or usefulness of Brams' models, but they certainly do affect the exposition in the book. For those of us who came of age, politically speaking, during the Reagan administration (or significantly later in the case of most of our students), the constant references to the primaries and elections of the Johnson and Nixon eras in a way that assumes the reader will know who the characters all are seems problematic. The discussion of impeachment has no mention of Bill Clinton or Monica Lewinsky and yet assumes a more intimate knowledge of the events surrounding Watergate than one can assume from students who have only learned of them from the history books.

Perhaps most strikingly, the book discusses third party candidates and their effect on elections without any reference to Ralph Nader, Ross Perot, or even John Anderson. One could argue that it is not an appropriate standard to hold a book to for all of the references to be completely up to date and that I am harping on this point too much. However, this reviewer feels that one of the joys in learning about the mathematics of voting and in discussing it with students is the immediate relevance to topics which are of current interest and losing this immediacy makes the models feel more abstract and less appealing to me and severely limits the usefullness of the book in a classroom setting.

I enjoyed reading *The Presidential Election Game*, and think that any reader will get quite a bit of food for thought out of the book — certainly there is more substantive discussion of elections here than on most of the 24-hour cable news channels. However, I wish that the author and publisher had done a more substantial revision of the book — perhaps in time for the 2012 campaign?

Darren Glass is assistant professor of mathematics at Gettysburg College