The mathematics of voting is more than just tallying the votes and the candidate getting most votes is the winner. Not that this book requires higher abstract algebra. In fact, a bit of combinatorics, the notion of a graph, the harmonic mean, and an order relation are the most advanced mathematical concepts that are used in this book and they are properly introduced when needed anyway. However, there are many different voting systems, and many criteria to define the winner. The problem is then, how to design the system in such a way that it is fair, which in turn requires to define what fair is. All this may lead to several definitions of the social model used and what kind of function is to be optimized. Hence theorems can be formulated and proved about which systems will define a winner an/or a loser and what social criterion is optimized. The proofs do not need much mathematics but they are mainly requiring precision and strict logic deduction.
Two chapters are devoted to two different kinds of voting. The first is a social choice model, and the second deals with yes-no voting. What holds for voting systems also holds for apportionment. In this book it is closely related to voting since it refers to the apportionment of the seats in the US House of Representatives that has to be decided every 10 years after the census or it may refer to the representation of different countries in international organizations. This book, is a textbook that is clearly addressing an audience of social science students, in particular in the US, although the general principles hold more generally of course.
In the first chapter, a society has to choose between two or more alternatives. A social choice procedure (or function) has to be designed that will define who is the winner or who are the tied winners and who are the losers. The problem is to define the choice procedure in such a way that a majority of the voters see their vote reflected in the result of the choice procedure. That is rather vague and thus leaves much freedom, and therefore many different possibilities, to organise the voting system and to define who are the winners and who are the losers. The chapter starts by explaining the difference between the plurality procedure (the group voting for the winner is the largest) or the majority (the group voting for the winner is larger than all the other groups together or larger that half the total number of votes).
Then, with this distinction in mind, the procedures can be complicated by organising several rounds, eliminating some candidates in every round, which may lead to a last round with only two candidates. Voters can perhaps give a ranking (like ranking a top three on the ballot with or without ordering them). Important is that the social choice procedure is monotone, which means that earning more votes should not turn a winner into a looser or conversely. This already gives many different systems, but when surveying the global "feelings" of the voting community towards the results, one may define a social welfare function which will define a ranking among (groups of) candidates. Important is that the relative ranking of two candidates by such a welfare function should be independent of the rest of the ranking. It should be independent of irrelevant alternatives (IIA). With all these restrictions, this approach to a voting system comes close to an axiomatic definition, which can have properties like neutrality, anonymity, or it can be dictatorial (i.e., where one voter or a group of voters can get a powerful dictatorial role). One has to change the axioms to turn the system from a dictatorial regime into an oligarchy and in such a way that it can not be manipulated. As proved by the many theorems in the text, it is difficult to find an ideal voting system.
In yes-no voting, the voter has only these two alternatives to vote: a yes or a no. This system is common practice when a candidate has to be selected for an important position or to accept or reject a motion or referendum. Here are fewer different procedures and hence the chapter is shorter than the previous one. Voters are grouped in coalitions. It now becomes important to define a power score of each individual voter. Since that depends on his/her position in the whole system of coalitions. Finding the power of a voter requires some combinatorial calculus. It even involves some probability (which is just a matter of counting) and magic squares (here only 3 x 3) to arrange the possibilities. It becomes more complicated when trades among coalitions are involved and when one needs to define the robustness of such a trade.
In the chapter on apportionment, the representation of a state can be proportional to it population, but it can only be represented by an integer number of persons. Thus some rounding (to the nearest integer) is required. Problems arise when the nearest integer is zero, or when the number of seats is larger than the number of groups to be represented (there are surplus seats to be distributed). Paradoxes can occur when a state looses a seat while its population has increased. Here again some monotonicity of the quota procedure should be imposed. Several criteria can be proposed, like for example looking at the per capita representation, that is the number of people in a state that are represented by each seat. One might for example minimize the difference so that each seat is representing (approximately) the same number of people. Other divisor procedures look at harmonic means, and there are many other possibilities.
The proofs of the theorems in the book are usually relatively simple, just relying on logical deduction rules applied to the definitions. It they are a bit more complicated, they are subdivided in a sequence of partial results. It should be noted that the author has definitely taken into account that his readership consists of non-mathematicians. So for a mathematical audience, the book could have used much more of the usual mathematical language and notation. Most of the text consists of examples that illustrate the possibilities of criteria that can be used and what result they give, which can be sometimes paradoxical. Each chapter has a list of exercises (answers to most of them are listed in an appendix). Thus the author has brought some mathematical rigour into a mainly non-mathematical subject, yet avoiding mathematical notation and formulation as much as possible, to transfer all this to non-mathematical students who would certainly not appreciate a more typical mathematical approach. El-Helaly has based this text on two decades of teaching experience. It is not only a textbook for his students, but it brings together a lot of material that is not easily found in this compact form and as such it will be of interest to any politician or anyone who is generally interested in the subject.