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

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. ;-)