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Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present Hardcover – 4 Apr 2010

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One of Choice's Outstanding Academic Titles for 2010
Honorable Mention for the 2010 PROSE Award in Mathematics, Association of American Publishers

"Mathematicians, economists, and political theorists have made their own attempts to elucidate the math of voting, and figure out better electoral systems. The story of these efforts is told in Numbers Rule. . . . Timely."--Anthony Gottlieb, New Yorker

"Clear and energetic. . . Szpiro charts this history selectively and with the use of major characters to render vivid a story of rival systems, which can easily degenerate into equations. He is a mathematician and uses tables to illustrate his arguments: but these are accessible to simple understanding. He is also a journalist and thus can tell a story."--John Lloyd, Financial Times

"Although voting problems manifest subtle mathematical complexities, Szpiro is an excellent communicator of mathematical concepts with a nimble ability to sidestep technical jargon. . . . An interesting, selective introduction into the complexities of voting reform."--Donald G. Saari, Times Higher Education

"A history of social choice theory, with much more detail (yet still readable) than one is used to receiving on this topic. I liked this book very much."--Tyler Cowen, Marginal Revolution

"I knew from reading Martin Gardner's columns that every voting system you can devise will occasionally turn up paradoxical results. . . . Szpiro walks you through the whole subject with very few equations."--John Derbyshire, National Review

"Engaging storytelling . . . 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."--Darren Glass, MAA Reviews

"In Numbers Rule, mathematician and journalist Szpiro presents a refreshingly different presentation of the mathematics of voting and apportionment. . . . The mathematical content is not trivial, and it is well written, very clear, and should be accessible to readers with an understanding of arithmetic and a willingness to play with numbers."--Choice

"Highly entertaining. . . . Anybody who has ever decried election results will be fascinated in Szpiro's accessible explanations of the paradoxes and enigmas that occur in all methods of election, from electing a pope in Rome, to apportionment of seats in the Congress by our founding fathers to ensure justice for all, even the minority."--Phil Semler, Sacramento Book Review>

"The author skillfully placed the development and evolution of the Social Choice theories in a broad historical context. The book shines in weaving the emergent math theories with historical circumstances. . . . [E]njoyable and informative."--Alexander Bogomolny, Cut the Knot

"Szpiro's book is a highly recommended good read on the history of the problems, which could illuminate a seminar series on the issues."--Ron Johnston, Environment and Planning

"It is an excellent addition to a growing body of literature that aims to convey ideas from the mathematical sciences to general audiences. Moreover, Szpiro's book is unique among other offerings in the mathematical social sciences in that it focuses on the historical development of the field. The narrative is engaging, witty, and easy to read."--Jonathan K. Hodge, Notices of the AMS

"George Szpiro's Numbers Rule does not break any new ground in the field of social choice theory, but it is probably the most entertaining book one is likely to find on the subject. . . . [A]ll social choice theorists should read this book. . . .Szpiro's book is ideally suited to be a supplementary reading for graduate classes in social choice theory."--Justin Buchler, Public Choice

From the Back Cover

"'Which candidate is the people's choice?' It's a simple question, and the answer is anything but. In Numbers Rule, George Szpiro tells the amazing story of the search for the fairest way of voting, deftly blending history, biography, and political skullduggery. Everyone interested in our too-fallible elections should read this book."--William Poundstone, author of Gaming the Vote: Why Elections Aren't Fair (and What We Can Do about It)

"Writing a book for a general audience on voting and electoral systems is a daunting task, but Szpiro succeeds admirably. He completely avoids technical jargon and focuses on the most important scholars and results in the field. This book fills a gap in the existing literature."--Hannu Nurmi, author of Voting Procedures under Uncertainty

"Numbers Rule focuses on key figures in the development of democracy and on the mathematics of voting, elections, and apportionment that they developed. Szpiro pays particular attention to the paradoxes that arise, and discusses them through examples."--Steven J. Brams, New York University

"Numbers Rule is very thoroughly researched and quite well written. The story Szpiro tells is both important and interesting. The most significant contribution this book makes is in the detailed history that it presents. It will have broad appeal."--Alan D. Taylor, coauthor of Mathematics and Politics

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7 of 7 people found the following review helpful
A Fun Study of the Mathematics of Democracy 10 Aug. 2010
By Eric Mayforth - Published on
Format: Hardcover Verified Purchase
In the United States, we now and then debate the merits of the Electoral College, but few people in this country pay attention to how the 435 congressional seats are apportioned to the 50 states after each decennial census. George Szpiro takes up this topic and other election-related problems in "Numbers Rule."

Szpiro describes how democracies from ancient Greece to the twenty-first century have dealt with the issues involved in making representation and elections as just as humanly possible. He describes how methods used to choose between multiple candidates progressed from those used to elect abbesses in the Middle Ages to those used in France in the eighteenth century, and shows the odd effects that can result when a third candidate is inserted into a previously two-man race.

This book was, appropriately enough, released in a year ending in '0', given that 2010 is a census year--the task of congressional apportionment will begin again soon. Szpiro recounts the intense debates between advocates of different apportionment methods in the early years of the republic and recalls many of the conflicts in later decades between states over the final representative apportioned. The author describes many of the mathematical issues that result, including the Alabama, New State, and Population Paradoxes--he shows mathematically how a state can, incredibly, lose a representative when the size of the House of Representatives is increased by one.

One trail that Szpiro did not go down involves the effect of an increase in the size of the House on presidential elections. Many people over the years have called for an increase of the size of the House of Representatives to anywhere from 600 to 1000 seats--in very rare instances this would be enough to change the result of an extremely close presidential election. Had the House contained, say, 870 seats instead of the 435 that it actually contained for the 2000 election, Al Gore would have won even without carrying Florida.

Szpiro reports the opinions of mathematicians concerning whether multi-candidate elections and congressional apportionments can ever be made completely fair, and provides brief biographical sketches of many of the mathematicians who dealt with these problems. The author closes by discussing election problems encountered in recent decades in Switzerland, France, and Israel.

"Numbers Rule" is a great study of the mechanics needed to put democracy in place and shows that they are not foolproof--one is reminded of Winston Churchill's assertion that "Democracy is the worst form of government, except for all those other forms that have been tried from time to time."
6 of 6 people found the following review helpful
Outstanding Historical Perspective on Voting 6 Sept. 2010
By Amazon Customer - Published on
Format: Hardcover Verified Purchase
This is a wonderful, historical look at voting systems. While voting might appear to be straightforward (we do it all the time), there are great difficulties in defining what the right winner is when there are more than two candidates. The standard "one with the most votes" (plurality) election is not particularly appealing since it is easy to split the opposition by adding candidates and having a candidate with very little support be the winner. This book traces understanding of these issues back to the Greeks and continues through the "dark ages", when clerics were particularly interested in how to elect Popes and abbots, through the Napoleonic period of Borda and Condorcet, up to the current era of impossibility theorems and computational complexity. The exposition is not mathematical (equations are exiled to chapter appendices) but it is analytical in the sense that concepts are clearly defined and the results are fully explained. Examples are interleaved to aid in understanding.

This book is highly readable and hits all the highlights. The exposition of the period from 1200-1450 was particularly interesting to me, since it is much less known than the relatively well-known French period.

If you have any interest in learning about why voting and apportionment are not straightforward, and want a readable, history-oriented book on approaches to these problems, I highly recommend this book.
5 of 5 people found the following review helpful
The Paradoxes of Democracy - A Search For Fair Choice 10 April 2011
By L. King - Published on
Format: Hardcover
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. ;-)
3 of 3 people found the following review helpful
Troubling Issues in Seeking Fairness in Elections 11 May 2011
By G. Poirier - Published on
Format: Hardcover
The intent of elections in a democratic society is normally to reflect the choice of a majority of the voting public. However, as is brilliantly shown by the author of this fascinating book, the reality of the matter can be far different and surprisingly complicated. In prose that is clear, friendly, lively and often quite captivating, the author guides the reader through the logic and mathematics (arithmetic, really) of the voting process through the ages. From the time of Plato to the present, we meet the many thinkers who have tried to make the voting process as fair as possible. The author clearly explains each of the proposed methods and both illustrates and lucidly compares them - their pros and cons - with useful examples, complete with relevant tables. As a bonus, a couple of chapters are devoted to the methods of fairly allocating seats in Parliament - another area that contains unexpected complications and surprises.

Although I found parts of the first chapter rather boring (i.e., analysis of Plato's views as translated from his writings), the pace rapidly picks up with each succeeding chapter, quickly making the book hard to put down. I found the inclusion of biographical appendices on several key individuals, as well as the couple of mathematical appendices, to very nicely complement the main text. This book can be enjoyed by anyone, especially those concerned about the fairness of our election process.
2 of 6 people found the following review helpful
The Text is Incompetently Formatted 17 Feb. 2012
By W.J. Walter - Published on
Format: Kindle Edition Verified Purchase
My comments do not concern the substance of this text but rather its presentation. The very first word of the book (on the first page of the Preface) is "It". Unfortunately, the upper-case 'I' is separated from the lower-case 't' by a blank line. This same problem recurs (throughout the book) whenever a word is italicized or presented as a link, and, consequently, the text appears ragged and sloppy. Since this is a problem that can be easily fixed, one can hope that the publisher will put out a corrected version and instruct those of us unfortunate enough to have purchased this disgraceful mess as to how we can download the corrections.

I should note that while I have verified that the problem I describe above appears in both the Kindle app for the PC and the Kindle app for the iPad, I do not know if it also shows up on the screens of devices made by Amazon.
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