# Mathematics for Democracy

There is mathematics in the New York Times today (December 6, 2015).  Not research-level mathematics, but math nonetheless.  Specifically, there is an article about using two simple statistical tests as indicators of gerrymandered voting districts.  By themselves, the tests don’t prove that something untoward has or has not gone on, but they provide a way of measuring bias.  The first, the difference between the mean and the median, is part of the Common Core standards for the sixth grade. The context of the article is an upcoming Supreme Court case on districting in Arizona.  Mathematical Reviews knows that mathematics has a lot to say about voting, including districting.  The Mathematics Subject Classification (MSC-2010) has a separate class about voting (91B12).  There is also the question of fair division, which is relevant to voting and political science, but also to other situations of resource allocations, such as radio frequency allocation, divorce settlements, or cutting a birthday cake.

A famous introduction to mathematics for the public is John Allen Paulos’s book, A Mathematician Reads the Newspaper, which is divided into sections by topics similar to the sections of most newspapers.  (A brief review of the new edition is available on MathSciNet.)  Paulos also wrote a book about Innumeracy.  The COMAP (Consortium for Mathematics and Its Applications) book, For All Practical Purposes, has a section on Social Choice (consisting of several chapters) that includes discussions of apportionment and of various voting methods.  The book is aimed at high school students and non-science undergraduates, but it is an excellent introduction for anyone.  You can pretend you are reading it in preparation for teaching a general math class for undergraduates.

Let me close with a few reviews of papers on voting or related topics.

MR3290334
McLean, Iain(4-OXNU)
Three apportionment problems, with applications to the United Kingdom. Voting power and procedures, 363–380,
Stud. Choice Welf., Springer, Cham, 2014.
91B32

This article provides an interesting overview of recent electoral changes in the United Kingdom that involve apportionment at several levels of government. The first section summarizes the theoretical and historical background on the problem of apportionment. The author contrasts three cases: (1) when a territory is divided into indivisible units and representatives must be assigned for each unit; (2) when members in multi-member districts in a party list system must be chosen under proportional representation; and (3) when members in a supranational body must be selected based on some criteria such as population size, etc. All three cases involve apportionment—the determining of an integer number of individuals to represent regions based on their relative proportions. However, differences in the underlying purpose of each situation give rise to different considerations. In case (1), for example, each territorial unit must have at least one representative so anapportionment method with a low threshold of representation might be desirable. In case (2), a higher threshold for representation might be preferable to minimize representation of splinter or extremist groups, and a method chosen accordingly. Finally, the desire to encourage or discourage coalitions among constituents may be important in cases (2) and (3); hence an apportionment method’s bias towards either large or small units is of interest.

The author reviews the best-known apportionment methods, using the history of Congressional apportionment in the U.S. to illuminate case (1), the divisor methods common to European proportional representation systems to discuss case (2), and the problem of assigning seats to the League of Nations and the European Union to analyze case (3). He discusses the properties of least remainder and divisor methods, identifies their “paradoxes” and notes the tradeoffs each entails with regard to bias, quota, monotonicity and threshold of representation.

In the second part of the paper, the author looks at recent electoral developments in the U.K. He summarizes the history of the allocation of seats to the House of Commons among the four countries in the U.K. (case (1)), which was traditionally determined through political solutions rather than adoption of a particular method, resulting in wide discrepancies among district sizes. After the treaty of 1921, for example, in which Northern Ireland remained as part of the U.K., its representation in the British House of Commons was kept low, since the Parliament of Northern Ireland held jurisdiction over many issues. In contrast, Scotland and Wales were overrepresented. Attempts in 1944 to create a system that would reflect dynamic changes in population resulted in a set of self-contradictory rules in which careless wording requiring minimizing differences in “electoral quota” (district size) rather than its reciprocal resulted in confusion between the methods based on the harmonic mean and on the arithmetic mean (Dean’s and Webster’s methods as they are known in the U.S.).

More recently, the U.K. has adopted a proportional representation system with multi-member districts for election of representatives to the E.U. (case (3)). A 1999 Act used a least remainder method (known as Hamilton’s method in the U.S.); in 2003, this was changed to the method of Sainte-Lague (Webster) by the Electoral Commission after consultation with a number of academic researchers. (The author includes, amusingly, excerpts from the 1999 debate in the House of Commons, where he is cited as having discovered mistakes in the calculations performed by the government in its research, under Home Secretary Jack Straw.) The article concludes with a discussion of the 2010 act in which the method of Sainte-Lague was adopted to assign seats to the countries in the U.K.

The article is well written and its organization is clear; the combination of theoretical discussion and analysis of British electoral apportionment changes should be of interest to those familiar with the literature, as well as those for whom it is new.
{For the entire collection see MR3290316.}

Reviewed by Jennifer M. Wilson

MR2382290 (2009b:91003)
Brams, Steven J.(1-NY-PL)
Mathematics and democracy.
Designing better voting and fair-division procedures. Princeton University Press, Princeton, NJ, 2008. xvi+373 pp. ISBN: 978-0-691-13321-8
91-02 (91B12)

What makes the modern world so different from the ancient? One major distinction is the prevalence of democratic institutions. In fact, it is now so well established that democracy is a general good that an invasion can be justified by claiming that it will “bring democracy”. The democratic peace principle, that democracies never fight each other, has been hailed as the closest thing to a scientific law that can be found in the social sciences [see, e.g., B. Russett, Grasping the democratic peace, Princeton Univ. Press, Princeton, NJ, 1994]. The effort to assess how democracy is related to national characteristics such as war-proneness has motivated many attempts to measure it precisely. For example, The Economist‘s widely-cited Democracy Index (available at http://www.economist.com/media/pdf/DEMOCRACY_INDEX_2007_v3.pdf) rates the level of democracy in 167 countries by combining measures of the electoral process and pluralism, the functioning of government, political participation, political culture, and civil liberties.

What exactly is democracy? Definitions differ, and there is no doubt that democracy comes in many varieties. Usually, countries are declared to be democracies when they develop systems of government and law that give citizens a say in the decisions affecting the country, and guarantee them some individual rights. Of course, there are many ways to implement government by the people, and many ways to ensure that citizens are treated fairly by their government. The book under review is a detailed study of some specific ways to address these problems; stated broadly, its objective is to find ways to strengthen democratic institutions.

Increasingly, mathematicians are finding interesting problems in social science, a development that the previous books of Steven J. Brams helped to catalyze. Mathematics and democracy, based on a selection of Brams’s (mostly co-authored) papers, will add to his influence. It concentrates on two procedural aspects of democracy: elections and fair division. It is obvious that elections and referenda are a central aspect of democracy. So, Brams argues, is fair division; it is, or should be, the basis for rules that determine how rewards and responsibilities are shared—by citizens, by regions, by those who are elected, and by their supporters. In summary, the major themes are:

• “how individual preferences can be aggregated to give a social choice or election outcome that reflects the interests of the electorate; and
• “how public and private goods can be divided in a way that respects due process and the rule of law.”

Inasmuch as democracy is procedure, an algorithmic approach is natural. Of course, some other politics-related problems had to be left out, even some that have a formal literature and can be seen as connected to democracy and fairness, such as districting, auctions, and matching.

Elections and referenda are the consultations necessary to achieve popular sovereignty, and the first half of the book is mostly about representative democracies, in which elections assign persons to positions. This is a procedural study, but it does not address rules about who can vote, what kinds of questions can be voted upon, or under what circumstances votes are cast. The main issues are exactly what voters are asked to indicate on their ballots, and how their ballots are aggregated to determine a winner. These details of a voting system determine, or reflect, something about the nature of a democracy. For example, if procedures are designed for proportional representation (voters vote for parties), there tend to be many parties, often tightly organized, whereas in first-past-the-post systems (each voter can support one and only one candidate), parties are usually less numerous and more loosely organized.

The most common single-winner election procedures ask voters to indicate their preferred candidate, to rank-order the candidates, or to indicate all acceptable candidates. Brams is an advocate of approval voting—a voter may vote for, or approve, any number of candidates; the winner is the most-approved candidate. Serious proposals for approval voting arose in the 1980s and subsequently. While there is little or no data on approval voting in public elections, there is considerable experience with it in the presidential elections of academic societies, including AMS, MAA, INFORMS, ASA, and IEEE. The adoption of approval voting by these societies is recounted, as is its subsequent rejection by IEEE. Evidence from these elections is reviewed to assess whether approval voting tends to “elect the lowest common denominator”, or “become ideological”. A second chapter on approval voting views its properties abstractly, and compares it to other electoral systems. Then several systems that change the ballot by allowing voters to indicate both approval and relative preference are described.

Single-winner procedures may not work well in multiple-winner elections, mainly because they often produce a set of winners that is not very representative. Several distinctive multiple-winner election procedures based on approval balloting (i.e., voters must approve or disapprove each candidate) are suggested; most of them aim for representativeness, but measured in different ways. For instance, one measures it on a category-by-category basis, while another produces a set of winners as similar as possible to the actual votes cast. Multiple (simultaneous) elections can bring another set of problems if voters’ preferences in one election depend on who wins another; this problem arises even in simultaneous referenda on related issues.

Fair-division procedures enter into democracy because they are needed to protect individual rights. For the most part, rights and freedoms are guarantees of fairness to individuals, guarantees that are hard to implement when they involve the allocation of something desirable and scarce. Majorities cannot be relied upon to decide fair allocations, because of the risk they will leave minorities with nothing. In other words, individual rights, one component of democracy, could be overwhelmed by the other, popular sovereignty.

Yet elections and fair division are not completely exclusive. In a parliamentary system, issues of fair division can arise after elections, when one party tries to assemble a governing coalition if no party has gained a majority. Cabinet ministries may then be allocated within the coalition, in accordance with a party’s number of seats and preferences. Like other indivisible goods, it is generally impossible to allocate cabinet ministries so that several reasonable criteria of fairness are satisfied simultaneously.

Two chapters address division of a single good; the problem is called divide-the-dollar if the good is homogeneous, and cake-cutting if the good is heterogeneous. These chapters surround a chapter on allocating multiple homogeneous goods, and are followed by one on allocation of indivisible goods in the presence of a single divisible good, which can be taken to be money. The recommended procedure in fact includes auction-like bidding on the indivisible goods.

The progression through voting procedures up to allocation procedures is summarized in a final chapter, which reiterates the argument that selecting voting and fair-division procedures, and improving them, depend on mathematical analysis. Ranges of procedures for voting and fair allocation are needed because there are many specific problems to be solved, and systems that do a good job on one problem may not do well on others. Yet the mathematics required for the analysis of these procedures is often not deep, and can be well suited to classroom use.

If democracy is important, then it is important to discover ways to make it work better. In Mathematics and democracy, Brams shows that comparing democratic procedures, and finding ways to improve them, is largely a mathematical enterprise. In the future, mathematicians may well discover many more good mathematical questions concealed in the social sciences in general, and politics in particular.

Reviewed by D. Marc Kilgour

MR1822218 (2002g:91001)
Saari, Donald G.(1-CA3)
Chaotic elections! (English summary)
A mathematician looks at voting. American Mathematical Society, Providence, RI, 2001. xiv+159 pp. ISBN: 0-8218-2847-9
91-01 (91B12 91B14)

After 11 September, 2001, one may wonder whether studying voting procedures is more important than ever. I will argue it is. It should be noted, of course, that I received the review copy of this book in June 2001 and consequently the book was written before this. However, it was written after the U.S. 2000 Presidential election.

Amartya Sen has described how freedom and development are closely interwoven. His recent book entitled Development as freedom [Oxford Univ. Press, Oxford, 1999] is a persuasive introduction to some of his work, showing why we must consider “$\ldots$the extensive interconnections between political freedoms and the understanding and fulfillment of economic needs”. This is not a widespread view. The promotion of political freedoms and economic development at the same time must become the motto of the so-called free world, today more than ever. Political freedom, probably the most important freedom, is, of course, linked to voting. The main object of voting is to reflect in the best possible way the voters’ opinion. The purpose of Saari’s book is “to explain what can go wrong with elections and why”.

During the last decade or so, Donald Saari has published numerous papers on voting and social choice theory, culminating (at this time) with two papers which appeared in 2000 [Econom. Theory 15 (2000), no. 1, 1–53; MR1731508 (2001e:91063); Econom. Theory 15 (2000), no. 1, 55–102; MR1731509 (2001e:91064)]. These two papers (forming in fact a monograph of more than one hundred pages) are, according to the author’s preface, one of two events that are at the origin of this book. (Incidentally, reading this preface is probably better than reading this review to convince potential readers to read the whole book.) The other event was the 2000 U.S. Presidential election. What follows is a brief description of the book’s contents. First, although one of Saari’s objectives is to persuade mathematicians of the interest of the mathematics of social sciences, I must say that the level of the required mathematics is not too demanding, even for people like me who are not professional mathematicians. This book can easily be read by social scientists with, say, some mathematical maturity. Even without this mathematical maturity, the most technical parts can be skipped without losing the main thread of the thesis.

The first chapter is mainly about political power. It describes difficulties within the U.S. Presidential election system, with the crucial rôle played by “small” states. It briefly reviews power indices (Shapley-Shubik and Banzhaf). (People interested in further developments must read the remarkable book by D. S. Felsenthal and M. Machover [The measurement of voting power, Edward Elgar, Cheltenham, 1998; MR1761929 (2001h:91032)].) Several alternatives to the U.S. system are presented, the chapter ending with a version of Arrow’s theorem. Saari’s comments on this theorem are nicely related to the U.S. 2000 election.

The second chapter draws on the author’s research on scoring rules. This domain is one which will definitely be attached to Saari’s name. This work is, in my view, one of the major contributions in voting and social choice theory since this subject has existed as a scientific subject (say, since Borda and Condorcet at the end of the 18th century). A scoring or positional procedure is a rule that assigns points to candidates according to their rank in the voters’ preferences. With plurality, for instance, one point is assigned to a voter’s top candidate and 0 to every other. With the Borda rule, with $k$ candidates, $k-1$ points are assigned to the top candidate,$k-2$ to the candidate ranked second, $k-3$ to the third and so on until 0 to the last. The outcome ranking is obtained by adding the points received by each candidate and ranking them according to the total they received. The author describes “the surprisingly wide assortment of paradoxical outcomes which can occur just by using different voting methods”.

The basic theme of the next chapter is to understand what can happen when candidates withdraw. Some of the results included were discovered by using techniques borrowed from chaotic dynamics. But be reassured. This chapter is as easy to read as any other chapter and you need not have taken a graduate course in dynamical systems to understand everything.

Chapter 4 is about strategic voting. Of course, you can easily imagine that most of the voters who ranked Nader at the top of their preference ranked Gore second. Since they surely knew that Nader had no chance to win, voting strategically meant for them voting for Gore rather than for Nader so that Gore could eventually win. In France, a few years ago, a President who was an applied specialist in strategic voting asked people to vote strategically, calling this “le vote utile” (useful voting). It seems that in Florida, some voters ranking Nader at the top of their preference and Gore in second position forgot to vote usefully, to Bush’s final benefit.

Chapter 5 is a credo in favour of the Borda rule. Basing his analysis on properties of symmetry and what he calls reversal effects, Saari shows that the Borda rule can be viewed as the procedure which meets what the voters, in some sense, really want. He is not the only person to favour the Borda rule. For instance, a famous British philosopher, Michael Dummett, who also happens to be an expert on voting, is also in favour of this procedure.

The last chapter deals with other domains in which Saari’s methods can be fruitfully used. These include aggregation in nonparametric statistics, probability, apportionment of congressional seats according to the population of the constituencies and, in voting again, proportional representation (problems of rounding).
I hope that I have now convinced you that this book is compulsory reading for mathematicians (all mathematicians), social scientists (nearly all, including, of course, economists) and most cultured laymen. Social choice theorists (economists or others) who already know Saari’s work must also read it. We always learn a lot from him. Incidentally, Cambridge University Press has recently published another of Saari’s books, entitled [Cambridge Univ. Press, Cambridge, 2001]. Do not miss it either.

Reviewed by Maurice Salles