This is a survey the history of mathematics of voting. It begins 2500 years ago with Plato in Greece. The focus is on Western history (Europe and North America), so there could have be significant, independent, historical achievements in other parts of the world not mentioned in this book.

The author is a good writer (a mathematician and journalist, per the dust jacket bio) who explains abstract ideas clearly. His sense of humor is an immense benefit for making a very technical subject enjoyable to a general, nontechnical audience.

There are only a few equations used, so a reader who is averse to mathematics should not be hesitant to try this book. Simple examples are sometimes illustrated in a table, and that often helps to convey the idea without the use of math.

Voting is an important subject in democratic countries, and understanding the intricacies is worthwhile for anyone. I would recommend this book to everyone.

Chapter by chapter summaries:
1. The Anti-democrat (Plato, b. 427 BCE, Greek)
In one of his final books, Laws, Plato employs his usual dialog method to suggest a form a municipal government for a new community of the island of Crete. The dialog deals with both the organization of the government and the method of electing officials. Election procedures laid out for various branches of government. The procedures vary and often seem arbitrary, without explanation why. Perhaps these procedures were so familiar to Plato's contemporaries that explanations were not judged necessary, and he was only documenting the overall structure.

2. The Letter Writer (Pliny the Younger, b. 61 CE, Roman Empire)
In a letter to a friend, Pliny relates his experience as leader of the court in which the jurists had a choice of 3 verdicts: acquittal, banishment or death. Pliny favors acquittal, and is dismayed to find the groups favoring the other two to be combining forces. Pliny encounters jurists employing what is now called “strategic” voting. That is, a voter anticipates that their first choice candidate is too unpopular to win. However, by casting their vote for another candidate, the voter can at least block their least preferred candidate from winning. Interestingly, Pliny considered that this was unethical.

3. The Mystic (Ramon Llull, b. 1232, Spain)
A Medieval monk and philosopher who wrote on theology, philosophy, mathematics and other subjects. In three of his works, an election process is described which involves pair-wise comparison among all candidates. Some of his writing now only exists in the form of transcribed copies. So scholars are not sure they can interpret his meaning accurately. However, this is an early example of a mathematically rigorous voting procedure.

4. The Cardinal (Nikolaus Cusanus, b. 1401, Germany)
A German Cardinal who read Llull's works, and then devised a slightly different scheme. In place of pair-wise comparisons, each voter orders the candidates from least to most favored, and awards them points. The least favored gets one point, the next gets two points and so on until the most favored is awarded the highest points.

5. The Officer (Jean-Charles de Borda, b. 1733, France)
A French Military Officer, mathematician and engineer who re-invents Cusanus's procedure. As Cusanus's precedure had been forgotten for a time (until the early 20'th century), it is now termed the “Borda Count”. Szpiro provides some good simple examples to show how election outcomes can vary if the number of candidates is changed.

6. The Marquis (Marquis de Condorcet, b. 1743, France)
A French mathematician who re-discovers Llull's voting procedure (which also had been forgotten in Condorcet's day). Condorcet also described what is now called the “Condorcet Paradox”. Szpiro provides a simple example to illustrate "cycles" and the paradox. A cycle occurs when, for example, candidate A is preferred over candidate B who is preferred over candidate C, who is preferred over candidate A. Because this does not appear to be rational, Condorcet developed a method for breaking the cycles. However, identifying all the cycles within the voting results can be a very large task.

7. The Mathematician (Pierre-Simon Laplace, b. 1749, France)
A French mathematician who analyzed both Borda's and Condorcet's procedures. He favored Condorcet's, because Borda's was susceptible thought to be susceptible to “strategic voting”. That is, a voter may rank an inferior candidate above a superior candidate in order to increase the chance of winning for the voters preferred candidate. The strategy can fail, if all voters employed it, but the possibility of strategic voting being used disturbed both Condorcet and Laplace. Laplace also advocated that with any procedure, the winner also must be required to win more than 50% of the votes.

8. The Oxford Don (Charles Dodgson/Lewis Carroll, b. 1832, England)
A British mathematician extends Condorcet's procedure, apparently without being aware of the original work. He proposes a method to break cycles.

9. The Founding Fathers (various, USA)
The topic changes from voting to calculating the proper number of representatives (aka "apportionment"). For the number of state representatives in the US Congress, Hamilton, Jefferson, John Quincy Adams, Daniel Webster, and others propose formulas which will comply with the Constitution (1 representative per 30,000 citizens). And all formulas are capable of producing slightly different results that will offend somebody. The “Alabama Paradox”, “New State Paradox” and “”Population Paradox”, are described.

10. The Ivy Leaguers (Wilcox, b. 1861, Huntington, b. 1874, both USA)
Describes Cornell and Harvard professors (Wilcox and Huntington) efforts to find an apportionment formula agreeable to all States. Their debate is re-ignited with each new US Census, and so was extended over several decades. The “ad hominem” arguments used by Professor Huntington against his opponent Wilcox were sometimes humorous, but also demonstrated, sadly, that scholarly achievement does not imply emotional maturity.

11. The Pessimists (Robert Arrow, b. 1921, USA)
Discussion of Arrow's Theorem which asserts that there is no voting procedure that is guaranteed to be fair under all conditions. Some simple examples might have made this section clearer. Also discusses “strategic” or “manipulative” voting and suggests that all methods are susceptible to these tactics (Gibbard and Satterthwaite).

12. The Quotarians (Michel Balinski, b. 1932, Peyton Young, b. 1944)
Further analysis of apportionment formulas shows that all formulas have some defects, but the Webster formula may be best. Table 12.3 contains typos.

13. The Postmoderns (various)
Discussion of Apportionment in Switzerland and Israel, and an experimental voting scheme in France. The Single Transfer Vote is mentioned, but not explained very clearly. ( )

For a book published in 2010, this has a strong feeling of 2001 about it.

The reason for this is not hard to understand: The Bush/Gore election of 2000 was the first in more than a century in which the candidate who won a plurality did not also win the White House -- and, more to the point, it was the first modern presidential election in which election procedures were not sufficient to allow us to know which candidate should have won under the laws applying at the time. (That is, we do not know which candidate was actually preferred by the majority of Florida voters, because of problems in counting the vote.)

But although the 2000 election brought home the problem, it really wasn't news to voting theorists. Everyone with knowledge in the field knows that plurality elections are not guaranteed to elect the candidate who would make the general electorate most happy. The question is, can we do better?

The answer to this is complicated. There is a mathematical result -- Arrow's Theorem -- which proves that it is not possible to devise a system which will always work to elect the best-loved candidate. It is, however, possible to do better than plurality voting. (Indeed, other than choosing at random, or letting a dictator make the choice, it is almost impossible to do worse.)

This book looks at the various methods proposed over the years to do better, starting all the way back with Plato (whose proposal for fixing democracy, to be sure, was to get rid of democracy) and running right up to the late twentieth century. Most of the figures who pop up in this discussion are pretty obscure (Jean-Marie, Marquis de Condorcet; Walter F. Wilcox) -- but some you surely know, including notably Charles Dodgson, or Lewis Carroll, England's first great voting theorist (whose proposal was that there should be what amounts to "ranked choice" or "instant runoff" voting, but with each district electing multiple representatives rather than just one).

To try to hold together both the theorists and the theory, author Szpiro arranges his history into chapters, and throws in sidebars with biographies of the theorists, or even of their theories. It's a method that seems well-suited to the topic.

But, somehow, it never quite jells. Some of that, to be sure, is that Szpiro doesn't cover all voting methods. (My personal favorite, points voting, is largely ignored even though it is comparatively successful in avoiding the problems of Arrow's Theorem.) The book will teach you a lot about the problems of voting, and about the various ideas proposed to overcome these problems -- but it won't really give you an idea of what makes them more or less effective.

And, of course, it won't teach you how to convince politicians elected by plurality voting that they should adopt a system that actually elects the candidates the people want to elect. But then, no one has managed that particular problem to date.... ( )