Customer Reviews

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

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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."

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.

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. ;-)

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.