A common theme in many liberal arts mathematics courses is showing students how mathematics gets used in the real world. One of the best examples of this is the issue of apportionment. It is easily motivated, and students can (with very little guidance) come up with many of the standard methods of apportionment. Typically, then, we show them the limitations of the various approaches: the paradoxes and the Quota Rule. None of the methods we present seem to avoid the paradoxes and satisfy the Quota Rule. And that's when the shoe drops: there is no perfect apportionment method!

This fact, (usually) known as Balinski and Young's Impossibility Theorem, was proved by Michel L. Balinski and H. Peyton Young in 1980. In 1982, they published the first edition of *Fair Representation: Meeting the Ideal of One Man, One Vote*, which summarized many of the results of the mathematics of apportionment. In 2001, they published the second edition.

Essentially, there are two parts to the book. The first part consists of Chapters 1 through 12. In these 93 pages, the authors give an overview of apportionment by using the history of the United States Congress. Seven methods of apportionment are presented: the methods of Hamilton, Lowndes, Adams, Dean, Hill, Webster, and Jefferson. The material is very readable, as the authors use history to describe how the various methods worked and the problems that arose from them. Examples of the Alabama, New States, and Population paradoxes are provided from US Apportionments. (Although the authors typically pick out a subset of the states to make the picture easier to see; for example, the Alabama paradox is shown using the 1880 Census results with Alabama, Texas, and Illinois.) It is remarkable (and quite serendipitous) that all of the pertinent issues that arise within the mathematics of apportionment have occurred within the relatively brief lifespan of our country. The authors do bring up a few examples from other countries, but everything that can happen, it seems, has happened here in America. There isn't much in the way of mathematical theory in this part of the book. The only time that the authors really attempt to delve into the theory is when they give an overview of the notion of a divisor method. And their explanation is quite intuitive, and very accessible.

The second part of the book, the Appendices, is where the mathematics occurs. In Appendix A, the authors present the theory of apportionment. They focus specifically on the divisor methods since these are the "population monotone" methods. Much of this appendix is spent considering which of the divisor methods is the most "fair." One of the interesting results is the fact that, depending on how you measure fairness, any one of several divisor methods could be considered the most fair. Using different combinations of apportionment and population, one can construct different ways of comparing apportionments, and then use these to come up with a divisor method. (For example, the method which minimizes the "relative unfairness" of an apportionment is Hill's method, which is the method Congress currently uses to apportion the seats in the House of Representatives.) The authors consider the notion of bias (whether a method tends to favor states with larger populations over states with smaller populations) and determine that Webster's method is the only divisor method which is "pairwise unbiased on populations," "pairwise unbiased on apportionments," and an "unbiased proportional divisor method." And, though the divisor methods are the only methods which are population monotone (thereby allowing them to avoid the various paradoxes), none of them manage to "stay within quota" for **every** apportionment problem. However, Webster's method is the unique divisor method which is "near quota, interpreted absolutely or relatively." After reading much of the appendix, it is hard not to develop a preference for Webster's method. Although the authors do show that Jefferson's method is the unique method which is population monotone and encourages coalitions, while Adams' is the unique method which is population monotone and encourages schisms. The authors also have an Appendix B, which contains the census data from 1790 to 2000, and the apportionments which result from the methods of Adams, Dean, Hill, Webster, Jefferson, and Hamilton. This gives the reader a chance to see how the different methods benefit larger or smaller states.

The authors do a great job of taking the seemingly easy job of apportionment, showing how it can turn out to be considerably less-than-easy, and then developing the mathematics to prove just what apportionment methods can and can't do. While the mathematics is not terribly accessible to anyone without at least an undergraduate level understanding of mathematics, the twelve chapters which comprise the main part of the book should be accessible — and educational — to anyone.

Note: this book is available in both hardcover and paperback editions.

Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State University. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at vestal@missoriwestern.edu.