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The Mathematics of Elections and Voting

W. D. Wallis
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Almost five years ago, I was asked by the chair of the mathematics department at Iowa State University to revamp our two “quantitative literacy” courses, typically taken by students who wanted to fulfill a mathematics graduation requirement without having to be exposed to the horrors of calculus. One of these courses, Math 105, is entitled Introduction to Mathematical Ideas and was largely a potpourri of topics chosen at the discretion of the instructor. Since my objective was to make the course more interesting, “real world relevant”, and uniform from semester to semester, I set about looking for a unifying topic, meeting the first two criteria, around which to base the syllabus.

It didn’t take long to discover that there were several books (e.g., For all Practical Purposes by COMAP and Excursions in Modern Mathematics by Tannenbaum) that devoted a number of chapters to what might be called “mathematics of social choice”; these chapters addressed various issues in voting, as well as apportionment and fair division protocols. It turned out that even though these topics were completely elementary and required no prior mathematics background to understand (at least at a basic, computational level), they were also actually interesting — and, for the most part, rather new to me, since they were certainly not part of a standard undergraduate or graduate mathematics education during my student years.

This may be on the verge of changing, however. The importance of mathematical analysis in elections was recognized by the MAA in 2008, when it made “Math and Voting” the topic for that year’s Mathematical Awareness Month. In addition, not only do books like the ones cited above discuss this material at a level suitable for math-phobic students, but there are also several books that are a notch or two more advanced, devoted entirely to these topics, and which make some effort to discuss some of the theory underlying the ideas. These books do not demand any sophisticated mathematical knowledge on the part of the reader, but do require a willingness to grapple with ideas and work through arguments and proofs that can on occasion be reasonably intricate.

Examples of this latter type of book, which are suitable for “topics” courses for majors or honors seminars for non-majors, are Mathematics and Politics by Taylor and Pacelli, The Mathematics of Voting and Elections: A Hands-On Approach by Hodge and Klima, A Mathematical Look at Politics by Robinson and Ullman, and Mathematics of Social Choice: Voting, Compensation and Division by Borgers.

And, now, we have the book under review as well, which is a nicely written introductory account of some of these topics, but which is very slim (less than a hundred pages of text), narrowly focused, and which, if it does offer a semester’s worth of material (I’m not completely sure it does), does so without providing much “wiggle room” for an instructor.

Before elaborating on these comments, though, it seems advisable to briefly look at just how mathematics enters into the subject of voting. Let’s start with an election where only two candidates are running. (The “candidates” need not be human beings; they can be outcomes, as in “Should the math club use its accumulated funds to finance a pizza party, or buy books for the library?”) In the two-candidate case there is a perfectly obvious way to resolve the election: just declare the winner to be the candidate with the most votes. (We won’t worry about ties, which can be resolved by any of a number of tie-breaking mechanisms.) While this is indeed the most intuitively obvious method, it is worth noting that, from a mathematical standpoint, it is also the most satisfactory one: a result called May’s Theorem, proved in 1952, asserts that “majority rules” is the only method that satisfies four certain “fairness criteria” for two-person elections. (May’s Theorem is not proved in this book, but a precise statement and proof can be found in the previously-cited book by Robinson and Ullman.)

When there are three or more candidates, funny things can happen. If three people run for office and receive, respectively, 35, 33 and 32 percent of the vote, then the winner here is a person whom a majority of the voters didn’t want to elect. This being viewed as an unsatisfactory outcome, attention turned to other ways of holding an election, and the concept of “preference ballots”, where every voter hands in a ballot listing not just that person’s choice but instead a ranking of the candidates in terms of preferences, makes an appearance. The results of these ballots can then be listed in a profile, and then a nontrivial question arises: how do we translate this tabulation of votes into a winner?

Various methods have been proposed over the years, but all have some flaw or weakness. For example, the Borda Count method awards points to every candidate based on that candidate’s appearance in a ballot: 0 points for every last-place entry, 1 point for every next-to-last appearance, etc. The winner is the person with the most number of points. One can generate simple examples, however, to see that if this method is used, a person can win an actual majority (not just a plurality) of first-place votes, and still lose the election. So the Borda Count violates what we call the “majority criterion”, one of several “fairness criteria” that we think a good voting system should have.

Another fairness criteria that the Borda Count violates is Independence of Irrelevant Alternatives (IIA): if, relative to a particular collection of ballots, candidate A wins and B loses, and then some voters change their ballots, but nobody changes their minds as to the respective positions of A and B, then B cannot now be the winner. (In other words, this criterion says that we would prefer not to have “spoiler candidates” who, as Al Gore discovered when he ran against George Bush, can change the outcome of an election.)

Another possible fairness criterion is the Pareto criterion, which states that if every voter prefers A to B, then B cannot win. It is clear that Borda Count does satisfy this criterion. Nevertheless, the failure of Borda Count to satisfy the other two criteria raises a natural question: is there any voting system which satisfies all of them? Can we find, in essence, a “perfect” voting system? A famous theorem of Kenneth Arrow gives a negative answer to this question and in fact establishes that a voting method for more than two people that satisfies even Pareto and IIA must be a dictatorship: i.e., there is some voter whose first choice candidate wins, no matter what the other voters say. (Arrow received the Nobel Prize in economics for his work in this area.)

These issues, fleshed out in more detail, animate most of the first five chapters (or roughly sixty percent) of the book under review. In them, the author discusses a number of various methods of deciding an election as well as a number of fairness criteria, and provides a proof of Arrow’s theorem. In addition, the author addresses the issue of manipulating the vote by casting an insincere ballot; here again, there is a famous result, called the Gibbard-Satterthwaite theorem, that, loosely speaking, states that any voting system with three or more candidates that satisfies minimal fairness criteria must either be manipulable or a dictatorship. This is proved here as well.

(This is not, by the way, the first time the author has written a book discussing the mathematics of elections and voting; his prior text Mathematics in the Real World devotes two chapters to the subject, and the book now under review can be viewed as an expanded version of these chapters.)

The remaining three chapters of the book address related issues. First, the author discusses voting methods for elections in which more than one candidate is to be elected (the author calls these complex elections, as opposed to the simple elections that were the subject of the earlier chapters). The next chapter addresses alternatives to preference voting, such as approval voting (each voter simply votes approval or disapproval of a given candidate, and the candidate with the most votes wins) and range voting (each voter has a certain number of points that he or she can divide up among the candidates).

Finally, in the last chapter, the author looks at weighted voting. The situation here is that there are a group of voters who must vote on an issue, but not all voters have the same number of votes. This chapter looks at several mathematical ways of assessing the power of a voter in terms of the number of votes that voter has. (This is, of course, not as simple as just counting the number of votes; for example, if a measure requires 8 votes to pass, and voters A, B and C have 4, 3 and 1 votes, respectively, then even though voter A has four times as many votes as does voter C, A’s respective power is exactly the same as C’s: either one of them, by voting “no”, can derail the measure.) The book discusses two different approaches (one due to a lawyer, Martin Banzhaf, and the other due to Shapley and Shubik), for quantifying power in a weighted voting system.

The exposition throughout is clear but concise. Because terms are given precise definitions and theorems are proved, this is clearly not a text that is intended for students who are afraid of these things. In addition, the book does not have the usual trappings of a very elementary text (color diagrams, boxed comments, etc.) but instead is the sort of serious-looking book that we have come to expect from Springer. So, it seems clear that the author is not aiming for an audience like my Math 105 students, but is instead looking at either math majors or non-majors in a more sophisticated (e.g., honors seminar) course. As a text for such a course, this book certainly does merit serious consideration.

I do, however, have a few concerns, some bigger than others. Perhaps the most serious objection I have is to the fact that the book lacks any index at all, which I think is a major omission for any mathematics text, even one as slim as this one.

In addition, I thought the exercises are, by and large, somewhat uninspired. Many of them are of a purely computational nature (“who wins such-and-such an election if such-and-such voting method is used?”); comparatively few call for proofs. This seemed to me to be somewhat at odds with the level of the book. While the text was probably too sophisticated for my Math 105 class, quite a few of the exercises were not. And, speaking of the exercises, the preface states that “answers to the odd-numbered questions will be found at the end of the book” and that a “complete solutions manual is also available”; both of these statements, though, seem incorrect. The end of the book contains only about four pages of solutions to a very small fraction of the questions, and not necessarily the odd-numbered ones (of the 19 questions in chapter 2, only three are solved; of the 24 in chapter 3, only one solution appears); moreover, at least as of this writing, I could not find a solutions manual, or suggestion of one, on Springer’s webpage for the book, though perhaps one will be forthcoming.

On a more substantive level, the very small size of this book indicates that some topics have been omitted that might profitably have been included. I would have liked, for example, to have seen a more extended discussion of the electoral college method of electing a President. The electoral college raises a number of fascinating issues, both political and mathematical, that are discussed at a rather high level of sophistication in Belenky’s Understanding the Fundamentals of the U.S. Presidential Election System and at a more student-accessible level in the Robinson/Ullman text referred to earlier. In this book, however, the subject receives only passing mention.

Likewise, in the area of weighted voting, there are some interesting ideas that are not mentioned here; for example, suppose we have a yes-no voting system that is defined not in terms of weights (number of votes per person) but in terms of what coalitions are necessary to pass a measure. The question arises: can we assign specific weights (i.e., number of votes) to the various voters so that the system appears as a weighted voting system? It turns out (see the book by Taylor and Pacelli mentioned earlier) that there is a nice necessary and sufficient condition for this to occur, but that subject is also omitted here. In addition, there is an interesting connection between power indices and electoral voting for President; assuming a two-person race, the states can be viewed as voters, and their relative power (using the number of electoral votes for each state as the weights) can be estimated. One can also inquire as to the relative power of voters depending on what state they live in; does a voter in California, for example, have the same influence over a Presidential election as one in Iowa? These ideas are also discussed in the Robinson/Ullman text, but are not treated in this text.

Omitting topics is often understandable when a book is already fairly thick, but it seems puzzling when the book is as slim as this one is. The lack of an index, the disconnect between the preface and the book, and the omission of a number of topics whose presence would, if nothing else, have made this text more versatile, all combined to give me the impression of a book that seemed like it was rushed into print.

To summarize and conclude: this book is a nicely written, concise introduction to an interesting area of mathematics that is probably not as well known as it should be. My biggest complaint is that it is perhaps too concise — it would have benefited, I think, by the addition of about thirty or forty more pages, particularly if some of those pages constituted an index.

Mark Hunacek ( teaches mathematics at Iowa State University. 


2.Simple Elections I

3. Simple Elections II — Condorcet's Method

4. Fair Elections; Polls; Amendments

5. Arrow’s Theorem and the Gibbard-Satterthwaite Theorem

6. Complex Elections

7. Cardinal Systems

8. Weighted Voting