This is the latest in a number of books, written over the course of the past decade, that discuss the mathematical issues raised by elections, voting. and apportionment. For a discussion of a number of these issues, and a survey of some of the other books that address them, please refer to my review in this column of

*The Mathematics of Elections and Voting* by Wallis. It should be noted that two of the books cited in that review, the ones by Robinson and Ullman and by Hodge and Klima, have since appeared in new editions, so it seems clear that there is some continuing market for textbooks on this subject.

This is not surprising. The mathematical issues raised here are both interesting and topical, and the material can be taught at several levels. Almost no mathematical background is required to understand the basic ideas involved (which have also been discussed in popular magazines; see, for example, the article “Win or Lose” in the July 19, 2010 issue of The New Yorker), but, at the same time, even without advanced mathematics, one can prove substantial theorems (including, for example, The Arrow Impossibility Theorem; Kenneth Arrow won a Nobel prize in Economics for his work in this area). I have successfully taught this material (without proofs) to first year, math-phobic, students taking a required “quantitative literacy” course at ISU, to more sophisticated students in an honors seminar (this time proving the important results), and to math majors in talks to the ISU Math Club. The proofs do not require any kind of mathematics beyond, say, what one might learn in high school, but do require an ability to reason logically and follow detailed arguments.

The book now under review has three chapters, one each on the subjects of preference voting, voting power and apportionment. In more detail, Chapter 1 addresses the issue, discussed in the Wallis review, of what is the “best” way to conduct an election where three or more people (or issues) are on the ballot. If voters submit “preference ballots”, in which they rank their candidates, it is obvious that these ballots can be collated into a “preference profile” listing how many people voted for each possible permutation of candidates, but it figuring out what to do with this information is much less obvious. There are lots of possible methods for determining the winner in a situation like this: for example (just to name two), there is the Borda Count method, wherein points are assigned to the candidates based on their ranking in each ballot, or the Hare method, in which any candidate with a majority of first-place votes is declared the winner, and, if no such candidate exists, the candidate or candidates with the fewest number of first-place votes is removed, the votes retabulated, and the process repeated, These methods, and others, each have their own strengths and weaknesses. The question is what the “best” method is. One way to approach this question is to articulate various “fairness criteria” that a good voting system should have, and then determining how many of these fairness criteria are satisfied by each method. Arrow’s theorem, mentioned earlier, essentially states that there is no “perfect” system.

The next chapter, the shortest in the book, discusses “yes-no” voting. The mathematical issue here is the assignment of some quantitative measure of “power” in situations where people, voting yes or no on a measure, do not necessarily have one vote each. For a simple example, suppose that there are three voters (call them A, B and C) with 3, 2 and 1 votes, respectively; to pass a measure, 4 “yes” votes are necessary. It is clear that A has the most power in this system (he or she is the only voter who can block a measure by voting “no”); how might we quantity this? There are several methods in use, including one credited to Shapley and Shubik and another credited to Banzhaf. These methods, and mathematical issues related to them, are discussed in this chapter.

Finally, the third chapter addresses apportionment. There has been considerable discussion recently about using mathematics to detect improprieties in the drawing of Congressional districts (see, for example, the article “Measuring Political Gerrymandering” by Kristopher Tapp in the July 2019 issue of The American Mathematical Monthly), but that is not the focus of this chapter. Instead, the issue here is how to apportion the fixed number of seats in the House of Representatives to the various states, given the Constitutional command that the number of seats a state gets must be proportional to its population. The problem, of course, is that this process generally involves rounding off non-integer numbers, and the question then becomes: what method of rounding off should be used? (This issue, of course, applies to other situations as well, such as when a department chair must apportion a certain number of teaching assistants to various classes based on their enrollment. These other situations, however, do not have Constitutional dimension.) What’s interesting is that, although various methods have been proposed (many by some of the great figures of American history: Alexander Hamilton, Thomas Jefferson, etc.), all have some flaws inherent in them. By analogy with the Arrow Impossibility theorem, it can be proved, in a sense that the text makes precise, that there is no perfect apportionment system

This is a nicely written book, with clear explanations that are supported by a number of useful, fully worked out, examples. However, it should be noted that definitions and theorems are stated precisely, proofs are given, and mathematical symbolism is not avoided. Thus, this is likely not a text that is appropriate for the kind of math-phobic freshmen that are likely to largely comprise a “quantitative literacy” course of the type referred to earlier; a more appropriate audience would be math majors in a proof-based course who already have some experience in reading precisely stated definitions and statements of theorems, and a willingness to track through the details of a proof.

Each of the three chapters ends with a good supply of exercises, some of which are computational and some of which call for proofs. Brief solutions to a number of these (typically the computational ones) are collected in a seven-page section in the back of the book.

The only real nit I have to pick with this book is the exclusion of certain topics that seem to me to be a natural fit. Two such topics, in particular, stand out. For example, Chapter 1 is limited only to preference voting, and does not address alternatives such as approval voting. This method has some enthusiastic supporters (such as Steven Brams, who has discussed it extensively, including in another Springer book titled *Approval Voting*), raises some interesting mathematical questions, and has been used by both the AMS and MAA in elections. Likewise, Chapter 2 fails to discuss the electoral college method of electing a President, which (assuming two candidates are running) can be thought of as a “yes-no” voting system in which the “voters” are the states, each state having its own number of electoral votes. Obviously, California plays a far more significant role in electing a president than does, say, Wyoming, but can we quantify the power of California versus that of Wyoming? Issues like these are discussed, for example, in The Mathematics of Politics by Robinson and Ullman.

While I think these omissions are unfortunate, there is still a lot of interesting material in this book. Instructors teaching a very introductory course on the mathematics of voting and apportionment will likely not select this book as a text, but may nonetheless want to keep it close at hand as a source of examples and enrichment material. Instructors teaching a more theoretical, proof-based course intended for mathematics majors should definitely put this book on their shortlist of possible textbooks.