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Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present 0th Edition

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George G. Szpiro (Author)
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George G. Szpiro (Author)
4.4 out of 5 stars 5 customer reviews
ISBN-13: 978-0691139944
ISBN-10: 0691139946
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  • Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present
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Product Details

  • Hardcover: 240 pages
  • Publisher: Princeton University Press (April 4, 2010)
  • Language: English
  • ISBN-10: 0691139946
  • ISBN-13: 978-0691139944
  • Product Dimensions: 0.8 x 6.5 x 9.8 inches
  • Shipping Weight: 1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #1,599,260 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

7 of 7 people found the following review helpful
A Fun Study of the Mathematics of Democracy
By Eric Mayforth on August 9, 2010
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.Read more ›
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6 of 6 people found the following review helpful
The Paradoxes of Democracy - A Search For Fair Choice
By L. King on April 9, 2011
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.Read more ›
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6 of 6 people found the following review helpful
Outstanding Historical Perspective on Voting
By Michael Trick on September 6, 2010
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.
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3 of 3 people found the following review helpful
Troubling Issues in Seeking Fairness in Elections
By George Poirier on May 11, 2011
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.
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