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

George G. Szpiro
PRICE (HARDBACK) £18.95 ISBN 978-0-69-113994-4

The books starts with Plato, labelled ‘the anti-democrat’. In Plato’s Republic democracy is eschewed but thirty years later in the Laws he relents and allows for some officials to be elected, but with a bias against the poor and less educated. To this end, Plato advocates multistage procedures which are well elucidated by Szpiro. Next the Roman lawyer and writer Pliny the Younger notes the problems involved when a committee (specifically a jury deciding both verdict and sentence) has to choose between three courses of action, and none of the three commands an overall majority.

The third chapter takes us to the thirteenth century and Ramon Llull. What is remarkable is that Llull’s work on elections for church leaders, which was rediscovered only in 1959, would be considered radical even today. (He was also an innovative theologian who apparently advocated enlarging the Trinity to include Jesus’ mother Mary). The next subject is Cusanus, a fifteenth century German Cardinal who daringly tackles the method of electing the Holy Roman Emperor. His system is similar to what is later called the Borda count, or as the author describes it (not in a complimentary way) the method used for the Eurovision Song Contest. Cusanus’s work, like that of Llull, lay forgotten for hundreds of years. The author goes into considerable detail about how and when texts were rediscovered or methods reinvented.

The story moves on to the period of the French revolution and the contributions of Borda, Condorcet and Laplace. A more rigorous mathematical treatment is now applied. Condorcet’s Paradox is the first major limitative result. In Victorian England, Oxford don Charles Dodgson (aka Lewis Carroll) independently arrived at similar voting schemes but with some innovations. A convincing case is made that Dodgson had not read the works of either Borda or Condorcet.

Several chapters are devoted to the protracted issue of how seats in the US House of Representatives are apportioned to the various states. This had the potential to flare up after each decennial census. In 1929 and again in 1948 Congress asked the National Academy of Sciences (NAS) to decide which method should be used. NAS preferred the Huntington-Hill (or Harvard) method which is the one now in use. However in 1982 Balinski and Young proved firstly that no one method has all the ideal properties, but also that an alternative method (Willcox-Webster) is optimal in a certain sense. Ironically given the foregoing acrimony, elections to the US Congress use first-past-the-post with all its distortions; but this the author leaves unmentioned.

Although much of the discussion is US-based, there is a comparative study of the representation of the Swiss cantons. Similar issues arise in the allocation of seats to parties under list Proportional Representation (PR) systems, including the Euro-elections in Great Britain. The Swiss bi-proportional system (first used in Zurich Canton in 2006) awards seats proportionally with respect to both party and district, and is explained by analogy with a Sudoku puzzle! Single Transferable Vote (STV) is briefly mentioned but the author does not clearly distinguish between multi-member STV and its single-member special case Alternative Vote (AV).

Arrow’s Impossibility Theorem is well explained starting with the axioms and conditions for a ‘social welfare function’, using Euclidean geometry as a comparative system. There is a novel attempt to link Arrow’s conditions with the Axiom of Choice (page 172), possibly relevant if the decision space is vast and complex. The related Gibbard-Satterthwaite Theorem shows that no system eliminates strategic voting. This is a theme of the book going back to Pliny the Younger and begs the question, is the assumed distinction between sincere and strategic preferences actually valid?

Overall this text is historically rigorous and well researched. Looking at developments through the eyes of visionaries including statesmen and mathematicians makes it very readable and requires little specialist knowledge. Written for a general audience, it will appeal to anyone interested in the development of democracy and as background reading for students of public choice.

Francis McGonigal CMath MIMA
Birmingham City University
Mathematics Today August 2011

Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present can be purchased at