Social choice has emerged as a popular topic in application-oriented liberal arts mathematics texts. The typical unit on voting theory in such a course might introduce preference balloting, run through some voting methods and fairness criteria, and conclude with a statement of Arrow’s Theorem, which says roughly that no voting system can satisfy all fairness criteria all the time. Game theory, apportionment, and power are other standard topics.
It is a fun unit to teach because we can demonstrate the utility and practice of mathematical reasoning with little mathematical background. Students enjoy it because they understand the importance of elections and there are plenty of ways to connect the material to their own studies and experiences (however one may feel about the 2000 U.S. presidential election, it was undeniably a boon for this course). So why not spend the whole term on it? Robinson and Ullman’s book is designed as a textbook for such a course.
The book finds a nice compromise between formality and accessibility. The authors take care to build from examples, isolate what is important, and generalize into theorems. It is expected that the reader has only limited mathematical experience, so much effort is put toward making very clear what is and is not being said. For example, the book is very explicit about what the goals are and that mathematicians are not casual about words like “impossible.” This is a real strength of the book — the patient reader will discover not just the content, but how and why mathematics is written the way it is. This is, therefore, a book that must be read and not skimmed. There are no colorful text boxes with vocabulary words, biographies, and anecdotes. Students who demand approaching exercises by finding similar examples in the chapter and skipping the narrative will find great frustration here.
There are four units: Voting, Apportionment, Conflict (probability and game theory), and The Electoral College (weighted voting, Banzhaf power). The exercises that close each chapter are interesting and often quite challenging — there are few straightforward computational practice problems. Solutions to the odd-numbered exercises are in the back. No college-level mathematics is required, but there is a fair bit of algebra and notation that, while generally introduced and explained with care, will intimidate some students. Though appropriate for an introductory course, the text is challenging.
Topics are introduced and motivated thoughtfully. Definitions are clear, and the authors take the time to explain why they need to be with well-chosen examples. When the proofs come (and they do come), they are set up properly. The examples and discussion are grounded in the American system of government and a general familiarity with American politics and history are useful. The downside to this approach is that other possible applications of the broad ideas in the text get little attention (and, of course, international students might need some background and persistent reminders).
There is a deliberate effort throughout to expose students to topics in mathematical reasoning that flow naturally from the content. For example, binomial coefficients come up in calculating the probabilities of ties and other election scenarios, and the value of approximation is covered when Stirling’s formula is employed to understand these quantities. By grabbing a specific application, students have the opportunity to explore when an exact value is needed, when an approximation is acceptable, and why it is often more important to understand the asymptotics of a function rather than its particular values. These distinctions are notoriously tough to sell in isolation and the authors do well including them here. The general goal, however, is the precise articulation of problems, the process of solving them, and the creation and recognition of ironclad solutions (often a proof).
The book has plenty of uses other than as a textbook. Instructors teaching a broader liberal arts mathematics course could use it to add depth to these topics or craft supplemental readings and projects. Students of mathematics or politics will find independent study opportunities here, and mathematicians from other areas will find this an enjoyable introduction. This is a very thoughtfully written text that should be made available to anyone with an interest in learning or teaching this topic.
Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa.