In time for the current American election cycle, William Poundstone's Gaming the Vote deals with the mathematics of voting. Readers of Fortune’s Formula, Poundstone’s previous book, will find that the author repeats his method of telling a story through vignettes (including interviews with mathematicians, political scientists and reformers), historical anecdotes, and discussions of basic results. Poundstone’s book doesn’t have a single equation, but the author’s engaging style attempts to bring a mathematical topic to a wide audience.
The mathematics of voting tends to be either a special topics course or a portion of a “liberal arts” mathematics course. While Arrow’s Theorem, an impossibility theorem for voting systems, is typically included among the topics of such courses, it’s usually among the concluding topics. Poundstone introduces Kenneth Arrow early into his discussions. His interview with Arrow, including Arrow’s struggle to find an appropriate thesis topic, can serve as an inspiration to graduate students everywhere. Arrow’s Theorem serves as a theme for much of the book, which also includes a discussion of the standard voting methods.
Voting methods for an election in which there are two candidates are fairly straightforward, but the problem becomes interesting for elections in which there are three or more candidates. With the plurality method, in which the person who gets the most votes wins, elections can be determined by the “spoiler effect” and vote splitting. Poundstone illustrates the effects of vote splitting in the American political scene through a number of historical elections, including the 2000 Presidential election in which many believe that Ralph Nader served as a spoiler to Al Gore and enabled George W. Bush to win the election. Plurality voting is widely considered to be a terribly flawed method, susceptible to manipulation, spoiling effects and negative campaigning.
In addition to plurality voting, Poundstone discusses other voting methods including the Borda count, the Condorcet winner, approval voting, range voting and instant runoff voting. Poundstone presents the arguments of the major proponents of various alternative methods to plurality voting using his interviews with the proponents. These are Don Saari (Borda count), Steven Brams (approval voting), Warren D. Smith (range voting) and Rob Ritchie (instant runoff voting). Poundstone finds the arguments of Warren D. Smith, a student of John Conway, to be the most convincing. Interestingly, both the AMS and MAA use approval voting to elect candidates.
Overall, Gaming the Vote is an excellent exposition of the issues and methods of the mathematics of voting, including its relevance to the political systems of democracies. I particularly liked the author’s interviews with Arrow, Saari, and Brams, which conveyed the human side of scholars whose works I have read. Poundstone also includes an interesting chapter on the work on voting systems by Charles Dodgson (aka Lewis Carroll).
I do have a few quibbles with the author. For example, while illustrating some potential issues of the Condorcet method on page 226, Poundstone states of a Condorcet winner, “But he doesn’t deserve to win.” The rationale for this statement eludes me. It seems to this reviewer that the winning criteria that one adopts determine the winner, not whether a candidate “deserves to win.”
I also have concerns regarding Poundstone’s endorsement of range voting, in which each voter scores a candidate on a numerical scale, say 1–10, and the candidate with the highest average is the winner. One potential problem with range voting consists of how the average is determined. For example, a few voters might be wildly enthusiastic for a candidate who is unknown to the majority of voters. If most voters do not rate such a candidate, should the denominator of mean automatically exclude nonrespondors? If so, then elections could be determined by small groups of fanatics! Of course, there are also problems with scoring nonresponses as zeroes. Poundstone briefly mentions this issue, but doesn’t really address this problem. Also, as any introductory statistics textbook will note, the mean may be influenced by outliers. Wouldn’t the median serve as a more robust measure of center? Although range voting is an intriguing idea, it’s clear that Arrow’s Theorem for voting systems is still quite applicable.
While Gaming the Vote will not settle the debate over voting systems, the book does provide an interesting and very readable summary of the subject.
Paul Schuette (firstname.lastname@example.org ) is a former Associate Professor of Mathematics who is leaving the world of academe for federal employment as a mathematical statistician. He is the current chair of the MAA's Basic Library List (BLL) Committee.