How do you select a winner from a field of candidates?
How do you rank a field of candidates?
How do you share a divisible resource, like a cake, or an indivisible one, like a pet or a house?
In the past two decades, it has become increasingly clear that mathematics plays an essential role in answering these questions. In recent years, a number of books have been published that seek to explain this connection. There have been a number of books examining the mathematics of voting such as Saari’s Chaotic Elections! A Mathematician Looks at Voting and Basic Geometry of Voting, Brams’s Mathematics and Democracy, Hodge and Klima’s The Mathematics of Voting and Elections. For the lay reader, William Poundstone’s Gaming the Vote: Why Elections Aren’t Fair and George Szpiro’s Numbers Rule: The Vexing Mathematics of Democracy from Plato to the Present are recently published books that provide an introduction to both the mathematics, the mathematicians, and an engaging historical context for the subject.
Börgers’s Mathematics of Social Choice: Voting, Compensation, and Division is a valuable addition to the mathematical literature as it seeks to explain not just the mathematics of voting, but also the mathematics behind “fair compensation” and resource sharing (“cake-cutting”). The book is based on Börgers’s notes for a course at Tufts University for liberal arts majors on these topics. The book is very well-written and is aimed at undergraduates without any college-level mathematical background.
The book succeeds at showing how elementary level mathematics can be used to solve interesting non-mathematical questions. While the mathematics is elementary, Börgers does assume that “the reader is willing to make an effort to work through mathematical arguments which, which while elementary, are not always easy.” A mathematician reading or teaching from this book will be pleased to see the usual mathematical setup of definitions, lemma, theorems, and proofs: this is definitely a math book. However, for the most part, it achieves a level that most entering college students will be able to understand and profit from. There are some chapters that are too technical or expect a high level of mathematical maturity, but these can be safely omitted as there are 26 chapters in the book, divided into three sections, and there is more material here than can be covered in one semester.
The only comparable book at present is Brams and Pacelli’s Mathematics and Politics which deals with similar topics and is also aimed at a liberal arts audience seeking to understand the applications of mathematics to social choice questions. The two books cover similar material but have different choices of topics and are written in different styles. For example, in their discussion of the mathematics of voting, both books discuss the usual topics of Arrow’s theorem, Borda count, Condorcet’s criterion, and spoilers, but Börgers organizes his chapters with the aim of arguing that Schulze’s beatpath method is the preferred method of winner selection methods, amongst those considered in his book. Brams and Pacelli do not mention Schulze’s method, and instead include chapters on weighted voting systems and apportionment, two topics omitted by Börgers. As far as the stylistic differences, both books have theorems and proofs, but Brams and Pacelli write in a more informal style, with many more examples, while Börgers is more concise. Both books have their advantages and instructors considering teaching a liberal arts course on this topic would profit by considering each of them.
Tom Hagedorn is Professor of Mathematics at The College of New Jersey.