I wanted to read Numbers Rule ever since I spotted a review of it at Mike Trick’s blog (by the way a theorem by Mike Trick is mentioned in the book).

The book is a concise history of voting systems, be it a majority voting system that deals with electing a leader, or with the proper apportionment of parliament seats. While doing that, using high school mathematics, it explains the algorithms that are used to appoint seats in a parliament (narrating the story of how the US Congress decided on the same issue) and gives a brief biography of key persons in the history of voting systems. The book stars Plato, Pliny the younger, Ramon Llull, Cusanus, Borda, Condorset, Laplace, Lewis Carroll, Willcox, Hill, Huntington, Arrow, Gibbard, Satterthwaite, Balinski, Young, Bader and Ofer. Von Neumann also! And I am always fond of books that mention von Neumann.

Reading this book was both educating and amusing. I understood why for some votes Robert’s Rules requires a 2/3 majority, instead of the simple 50% plus one vote. Do you really think that a 50% + 1 is a selection that represents the will of the people? Think again. The book is full of examples of why this is not the case, including strategic voting. It was amusing enough for me while discussing fair apportionment methods because I always had in my mind the proponents of this magic voting system in Greece (which has never been put to the test) that allocates members of parliament is a fair manner. It struck me that for so many years the proponents always mention the fairness of the system but I have never ever seen the mathematics of it documented anywhere. And it is not that it is complex or complicated stuff. But hey, if you put it down on paper you have to deal with reality and the fact that a seat in the parliament cannot be fractional.

The book is heavily focused in the history of the apportionment of seats in the US Congress, but it also makes a brief passing from Eurovision (although the author makes a small error about the voting points) and also discusses the Swiss and Israeli methods of apportioning seats in a Parliament. The Israeli method of party alliances is really interesting. One should not be let down by the heavy US orientation of the book. It is used by the author as an excellent vehicle in order to present apportionment paradoxes.

Arrow’s paradox takes about a chapter in the book, and the author manages to explain to the layman what Arrow proved with his thesis (and with specific references to certain pages of it). If you want to have a quick explanation of why Arrow’s theorem is important start from this chapter.

In the end I liked this book, because it added two more books in my reading list: The Arrow Impossibility Theorem and Fair Representation.

A lot of people speak the word Democracy. Few understand it. Start testing your understanding by reading this book.