Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present

by George G. Szpiro

A lively history of the peculiar math of votingSince the very birth of democracy in ancient Greece, the simple act of voting has given rise to mathematical paradoxes that have puzzled some of the greatest philosophers, statesmen, and mathematicians. Numbers Rule traces the epic quest by these thinkers to create a more perfect democracy and adapt to the ever-changing demands that each new generation places on our democratic institutions.In a sweeping narrative that combines history, biography, and mathematics, George Szpiro details the fascinating lives and big ideas of great minds such as Plato, Pliny the Younger, Ramon Llull, Pierre Simon Laplace, Thomas Jefferson, Alexander Hamilton, John von Neumann, and Kenneth Arrow, among many others. Each chapter in this riveting book tells the story of one or more of these visionaries and the problem they sought to overcome, like the Marquis de Condorcet, the eighteenth-century French nobleman who demonstrated that a majority vote in an election might not necessarily result in a clear winner. Szpiro takes readers from ancient Greece and Rome to medieval Europe, from the founding of the American republic and the French Revolution to today's high-stakes elective politics. He explains how mathematical paradoxes and enigmas can crop up in virtually any voting arena, from electing a class president, a pope, or prime minister to the apportionment of seats in Congress.Numbers Rule describes the trials and triumphs of the thinkers down through the ages who have dared the odds in pursuit of a just and equitable democracy.

Genres: Science, Nonfiction, Politics, History, Mathematics, Society, Philosophy

235 pages, Kindle Edition

First published March 1, 2010

About the author

George G. Szpiro is an Israeli-Swiss applied mathematician and journalist, who has emerged as a writer of popular science books.

Reviews

[Davide · Rating 3 out of 5]

Un buon libro per capire qualcosa in più su come i voti nei sistemi democratici si possano tradurre in seggi parlamentari, fra contraddizioni, sproporzioni e stranezze storiche che possono far sorridere

[Phoenix · Rating 4 out of 5]

The Paradoxes of Democracy

This is a book about the mathematics of elections and the implications of different methods towards fairness. The progression of topics is both historical and biographical. Szpiro begins with an interesting discussion of Plato and examines the principles and merits behind the scheme proposed by "The Athenian Stranger" (a stand-in for Plato himself) in "The Laws". The presentation here was much more interesting than what I recall being taught, however the scheme is entirely academic - it was too rigid and idealistic to ever be implemented.

The book continues with the prolific Greek writer of letters Pliny the Younger and two interesting problems that came up in his career - that of fair representation (about a lawyer who failed to appear for his clients) and the unfairness of strategic voting where the assembly had to choose between three options in a murder trial - a plurality of 40% favouring acquittal, but 30% favoring the death penalty and 30% favoring banishment.

Szpiro's next stop is the late 13th century with two schemes proposed Raimondo Llull. All are forms of weighted voting. The first involves time consuming pairwise comparisons of each of n candidates where the winner winds the most matches. He later modifies the technique to handle ties. Lastly Szpiro examines a later manuscript which applies a round robin pairing to determine the winner. The method is flawed as it favors candidates who are considered later on and because preferences are not measured consistently on a single attribute and therefore choices are not transitive. In other words, If I prefer Jim to Mark and Mark to Sally it does not follow that I prefer Jim to Sally. Jim may have a better foreign policy than Mark. Mark may have a better domestic policy than Sally but Sally may be more competent and able to implement policy than Jim.

Llull's work is picked up in modfified form in the 15th century by Cardinal Nikolaus Cusanus for eclesiastic elections. Here the voters are given slips of paper marked 1 to n where they rank each of the candidates. The candidate with the lowest score is the winner. Szpiro goes on to describe a modern variation that is used in the EuroVision song contest.

Fairness and lack of transitivity in choice during les temps dangereuse of the French Revolution, are the themes in Chapters 5-7. Jean-Charles de Borda and the Marquis de Condorcet both propose competing schemes similar to that of Llull and Cusanus, with Condorcet favoring two-by-two contests and providing an analysis of the problems with Borda's technique and Borda favoring weighting. The third proposal is that of mathematician LaPlace who favored a series of runoffs - the same system that is used in France today to select the President.

Chapter 8 is an interesting segue on Englishman Lewis Caroll and his analysis of proper voting which consists of an independent rediscovery of Condorcet's approach. He applied it to a vote on hiring a colleague and the selection of an architectural design for a new building at the college.

The next 80 pages (Ch 9-12) concentrated on the American Congressional System and by extension the Electoral College which elects the US President. Here the issue is allocating a fair number of seats to each state where the number of seats depends on the size of the population. The problem is that the number of seats has to be an integer and the seats are localized to each state. If there are 400,000 voters per seat across the Union and 900,000 people in Montana then Montana gets 2 seats not 2.25 seats. 100,000 people in Montana are underrepresented. The suggestion that a 3rd representative be sent to Congress who's vote counts only for .25 is briefly suggested but its not analyzed to any degree. Instead Szpiro looks at 5 alternate proposals that involve rounding either up or down. Szpiro helps us follow the political debate. None of the solutions are completely "fair" and all lead to potential paradoxes, some favoring large states, some favoring small one, but there's a new twist - the analysts now turn to measuring the degree of fairness. Since the allocation of seats follows a census, and the census was just last year, the debate as to which method to choose may become current again.

The final chapter looked at foreign jurisdictions - Switzerland which uses a complex scheme where excess votes in one canton can spill over to another, and Israel which uses a proportional voting approach and added the innovation that prior to the election similarly principled parties can openly agree to assign votes not used to elect a candidate in their party to the other party. Both ideas address the American problem of people being reluctant to vote because their vote for an unpopular candidate or cause is thought to be wasted.

I really enjoyed the conversational tone and the clear explanations given both to the methodology and the flaws in each technique. The biographical side notes at the end of each chapter were for the most part interesting, though the discourse on Pliny and Vesuvius a bit long, and in some other cases simply there for consistency of format and could be skipped. I wasn't too happy with the description of Arrow's Axioms which I thought needed more coverage. (I did study Arrow's proofs in University, so my expectations may be probably higher than most.) Szpiro could also have written about bicameral systems with upper and lower houses, cases where more than a simple majority is required (ie: carrying an amendment) or cases where veto power exists. I also felt that the notion that there are other factors than purely numeric superiority which can weight the vote should have been looked at - for example Lebanon which is a confessional system, Belgium which balances Flemish vs Walloons or Canada which tries to add balance to different regions in effect giving land a voice at the table. I also thought it would have been interesting to examine the cases of Italy and the UN as well as the power of subcommittees to frame agendas for the whole. Szpiro did touch on this a number of times but I felt it needed to be tackled more fully - though numbers may rule, the power to frame the question may contribute more to the answer.

In summary: I enjoyed the book as far as it went but I'd have like seen a bit more. I'd give it a fractional rating of 4.1. ;-)

[Caleb · Rating 3 out of 5]

Several chapters from this book were assigned reading in my course on Social Choice. I enjoyed those selections, so I decided to finish the remaining chapters. I find Szpiro's writing to be quite enjoyable despite the fact that the topic is predominantly mathematics. By adding the intrigue of politics, Szpiro is able to add many interesting anecdotes and tongue-in-cheek comments that make the reader chuckle. His particular pet-peeve seems to be the Bush-Gore election (it's given several mentions as proof of how voting systems, democratic/majoritarian/what-have-you, are always flawed because of the human factor). Overall, I really enjoyed the book. Szpiro is clear to point out that he glosses over the majority of the mathematics in order to relay the important and impossible points, but knowing a bit about voting methods and Arrow's theorem allows for a more enjoyable read.

[Darrenglass · Rating 5 out of 5]

[As always, I dont review books here that I read for work, except when I do. This is one of those exceptions. My full review can be found online at http://mathdl.maa.org/mathDL/19/?pa=r...]

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Given that individual members of society have their preferences, how should we aggregate them to figure out what society as a whole prefers?

How should we divide financial resources, pieces of cake, or congressional representation between people in a way that is as fair as possible?

These questions and similar questions related to social choice have many possible answers, and have been thought about for a long time by philosophers, political scientists, pundits, and, yes, mathematicians, with different groups each having their own take on the issues. The mathematics of voting is a subject that has been very well-tread in recent years, with books by mathematicians such as Brams, Straffin, Taylor, and (probably most famously) Donald Saari being just some of the books in the field that have been reviewed by your team of MAA Reviewers. Unsurprisingly, most of these books focus on the mathematics behind the mathematics of voting. A new book by George Szpiro, entitled Numbers Rule: The Vexing Mathematics of Democracy from Plato to the Present is also about the mathematics of voting and social choice, but takes a different approach, focussing on the history and the people who have brought us various mathematical developments in the field.

Each chapter of Szpiro’s book focusses on the contributions of one person (or, in a few cases, a small group of people) to social choice theory. The first set of chapters deals with voting, and how we combine individual preferences to figure out what preferences society as a whole may have. The book begins with a chapter entitled “The Anti-Democrat” about Plato’s idealized city, as described in The Republic and Laws, in which traditional democracy is replaced by an elaborate system of guardians and magistrates and Cnossians. Another chapter, “The Letter Writer”, looks at letters of Pliny the Younger, who was concerned that the outcome of a vote between three options did not best represent the will of the people, predicting some of the problems that third party candidates face in contemporary US Elections.

Later chapters (“The Officer” and “The Marquis”) look at people whose names will be familiar to anyone who has done reading on voting theory, Borda (of “Borda Count” fame) and Condorcet. While the mathematics discussed in these chapters is unlikely to be new or surprising to MAA members who read the book, Szpiro discusses many biographical details which may be. For example, did you know that Condorcet was a feminist who, despite feeling that women should not have the right to vote, did ask “Why should beings exposed to pregnancies and to passing indispositions not be able to exercise rights that no one ever imagined taking away from people who have gout every winter or who easily catch colds?”

Later chapters in the book shift away from the questions of voting and move to questions of congressional representation. In particular, how many congressmen should each state receive? The US Constitution is surprisingly vague on this question, specifying only that each state shall have at least one representative and that “the number of representatives shall not exceed one for every thirty thousand.” It is left up to congress to decide how many representatives there are in total, and how many should represent each state. Unsurprisingly, this vagueness has led to many conflicts over time, and in Numbers Rule, Szpiro discusses the various methods proposed by founding fathers such as Webster and Hamilton, as well as “The Ivy Leaguers” Walter Willcox, Joseph Hill, and Edward Huntington. Szpiro leads us through a series of congressional debates, special panels, and politicking, explaining along the way the mathematical problems and “paradoxes” that the various solutions lead to. These chapters have more numbers than the chapters on voting, and are slightly more technical, although the emphasis is firmly on the people and the storytelling.

The question of how we vote and the question of how to apportion seats in congress are both quite interesting and both have unsatisfying endings. As most mathematicians know, Kenneth Arrow proved in 1949 that there is no system of voting which satisfies a small number of conditions that just about anyone would want from a voting system. Similarly, Allan Gibbard and Mark Satterthwaite showed that any election can be manipulated if some voters misrepresent their preferences. More recently, Michel Balinski and H. Peyton Young have shown that there is no system of congressional apportionment which is both unbiased and avoids the so-called “Population Paradox” in which states lose seats despite a growing population. Szpiro discusses all of these theorems and how they muddy the waters and remove any hope of a wholly satisfying solution to the questions we started with. A final chapter, entitled “The Postmoderns” considers several systems that have been developed in light of these theorems, such as Switzerland’s model of apportionment and the single transferable vote model which is used in house elections in Australia as well as many city council elections here in the United States.

One nice feature of Szpiro’s book is that each chapter ends with a biographical appendix, typically a page or two describing aspects of the lives of the people he is writing about that do not pertain to social choice theory. Several chapters also contain mathematical appendices giving proofs of the more technical results discussed in the main text. These features, as well as Szpiro’s engaging storytelling, make for a very nice book. There are certainly more rigorous and in-depth mathematical treatments of the issues that Szpiro discusses, but for a reader who is primarily interested in learning some of the historical context of the characters who have contributed to the mathematics of social choice theory, it is hard to imagine a better book.

[Simone Scardapane · Rating 4 out of 5]

Il libro è interessante ed ogni capitolo copre uno o più personaggi ed aspetti legati ai sistemi di voto, con un forte focus sulla ripartizione dei seggi in un sistema federale come gli Stati Uniti, ma spaziando nel resto del libro da Platone ad Arrow. Si legge piacevolmente a meno delle note di appendice, che spezzano il filo del discorso in diversi punti (in alcuni casi con cinque o sei biografie di fila). Purtroppo (almeno nella mia edizione) c'erano anche alcuni errori tecnici, come il termine esponenziale riferito ai problemi di tipo P oppure il fatto che le geometrie non Euclidee abbiano meno assiomi di quella Euclidea. Nonostante questo una piacevole lettura su un tema molto di nicchia.

[Offbeat Reader · Rating 4 out of 5]

It's an interesting book. I picked a chapter everyday as it has a lot of factual data with some interesting points. Narrating the historical development of democracy.

I enjoyed the book and background stories. It shows how laws and perception of today's world have evolved. It is much more relaxed and promotes equality. But the common theme is the rich has always benefitted from these laws. There were some references to socialistic theories that were developed later which used some of the common themes from Plato's book.

Recommended if you are curious.

[Gabriel · Rating 3 out of 5]

As a math major who attended a college known for its political sciences, this book was right up my alley. But even more pertinently, I once devised a voting scheme to help my a cappella group select several songs. As it turns out, my method was similar to a system halfway between that of the medieval scholar Ramon Llull and a method by the famous 19th century author (and mathematician) Lewis Carroll. Reading this book years later, I learned that the system I had concocted was unfortunately susceptible to “strategic voting,” and that a minor modification could have improved it (although, I am proud to say that in practice it worked quite well). Oh well.

So was I fascinated by this book? You bet. Szpiro writes approachable, engaging explanations of the truly “vexing” problem of making group decisions, demonstrating the numerous procedural and theoretical pitfalls of voting schemes and issues of parliamentary / senatorial representation. Anyone remotely interested in mathematical history, politics, or logic in general should enjoy this book.

I did find that certain points were repeated a little too often, even within a single chapter. As a mathematics enthusiast I would have liked to see more of the referenced proofs actually included in the appendices (not just the short simple ones). And some of the most interesting personal histories of the relevant people were confined to end-of-chapter text boxes; that material could have enriched the main narrative even if it wasn’t strictly about the central topic of voting systems.

But really I can recommend this book as both an interesting and enlightening look at a problem which effects *everyone* who lives in a democracy or participates in any kind of group decision. This material isn’t abstract; as the title so eloquently phrases it, numbers really do rule us, and it is important to understand how and why.

[Maurizio Codogno · Rating 5 out of 5]

Forse avete sentito parlare del teorema di Arrow, che dimostra matematicamente come - se vogliamo scegliere dei rappresentanti secondo un certo numero di regole assolutamente sensate - l'unica opzione possibile è avere un dittatore. Magari avete anche sentito parlare del paradosso dell'Alabama: suddividendo per stato in modo proporzionale i rappresentanti degli stati USA al Congresso e aumentando il numero di rappresentanti totali, era possibile che uno stato perdesse un seggio. Ma ci sono molte altre cose da sapere, e George Szpiro ce ne racconta davvero tante in questo libro. Si può scoprire per esempio come anche Lewis Carroll abbia cercato di risolvere il problema del voto, che il primo a pensarci seriamente è stato nientemeno che Platone e che anche Plinio il Giovane aveva tentato di fare qualcosa al riguardo, e che negli USA ci sono almeno cinque modi leggermente diversi di decidere come dividere i seggi al Congresso per stato, e le scelte sono spesso molto più politiche che matematiche. Naturalmente viene spiegato cosa dice esattamente il teorema di Arrow, e in più si aggiunge il teorema di Gibbard-Satterthwaite che dimostra sempre matematicamente come le elezioni possano essere manipolate.
La matematica presente nel libro consiste al più di divisioni, quindi anche i matematofobi possono leggerlo; per chi non sa l'inglese segnalo la traduzione italiana appena uscita per i tipi di Bollati Boringhieri, anche se non posso garantire sulla qualità della traduzione avendo io letto il testo originale, del resto molto scorrevole.

[Ralph Orr · Rating 4 out of 5]

A good introduction to the problem of fair proportional representation in democratic systems. The reader needs some comfort level with math.It explores the background to and the history behind how the United States apportions the House of Representatives in close conformity to the requirements of the Constitution, as as well as the impossibility of removing all ambiguities and weaknesses thereof. Beware three-or-more candidate elections. They are most prone to manipulation.

[Andrew · Rating 2 out of 5]

I generally like books on this topic, but this one missed the mark. The author spends at least as much time discussing the personal lives of the mathematicians and thinkers that came up with different voting schemes as he does discussing the "numbers", as advertised in the title. The stories themselves aren't very engaging, and the book failed to keep my attention and interest.

[Frank · Rating 3 out of 5]

Not what I was expecting: we find here the tale of Pliny's account of the eruption of Vesuvius and read about the medieval romances of Ramon Llull. Decision theory is touched upon in the most tangential fashion. Not that I mind: Szpiro's narratives are interesting and amusing, but the lead word in the title is hardly merited.

[David · Rating 3 out of 5]

Um livro sobre um tema muito interessante, fundamental para qualquer entendimento do processo democrático, mas que sofre pelo estilo demasiado pragmático do autor, principalmente os últimos capítulos o de o texto não está tão bem conseguido.

[Ami Iida · Rating 2 out of 5]

This book describes a decent respect to election contents.
It applies for the game theory.
But sentence wrote lazy.
I want it to edit more.