Decisions and Elections: Explaining the Unexpected

by Donald G. Saari

This highly accessible book offers undergraduates and professionals a new, different interpretation and resolution of Arrow's and Sen's theorems. Using simple mathematics, it shows that these negative conclusions arise because, in each case, some of their assumptions negate other crucial assumptions. Once this is understood, not only do the conclusions become expected, but a wide class of other phenomena can also be anticipated. These include inter alia legislative cycles, supply and demand economics, statistical paradoxes, and diverse voting/election paradoxes.

256 pages, Hardcover

First published October 22, 2001

Reviews

[Fernando Pestana da Costa · Rating 5 out of 5]

Donald Saari is a renowned mathematician, former Editor-in-chief of the Bulletin of the American Mathematical Society, and author of important work in two paradoxically distinct subject areas: dynamical systems (mainly n-body problems in Newtonian mechanics) and the mathematics of voting systems, where he is arguably the current foremost world authority. He is also a prolific popularizer of this last topic, having to his credit (up to now) the authorship of three books with varying degrees of mathematical pre-requisites (a fourth is announced for the fall of 2008). The book under review is the least mathematically demanding of them. It is an excellent place to start learning about voting and decision-making procedures and the host of unexpected outcomes, some really apparently paradoxical, that can occur. The book explains in very clear and simple terms the hypothesis and context of Arrow's and Sen's celebrated theorems, then, along three chapters, it exemplifies and explores what is the reason underlying Arrow's, Sen's, and maybe other similar results: the inability of much voting and choice procedures to use the connecting information between the parts and the whose, and the concomitant inability to distinguish between rational and irrational voters. Finally, Saari shows a resolution out of the problem in Arrow's theorem by introducing the notion of intensity of pairwise ranking between alternatives, with which Saari proved (elsewhere) that the Borda Count is a nondictatorial procedure satisfying the analogous Arrow's type conditions. This is an extremely interesting book, with close to nil formal mathematics, but that should be read by everyone interested in the subject (be him a mathematician or otherwise) for its clarity of exposition and the capacity of Saari to explain fine points and difficult problems and results in a transparent way and with a minimal amount of technical requirements.

[Willis · Rating 3 out of 5]

Word of warning - this is a technical book written by a mathematician. If you don't have much background in mathematics, you will likely find this boring and difficult to follow.

Saari describes why no voting method works universally well but then goes on to give some conditions that help overcome the paradoxes of Arrow's impossibility theorem. If you are not familiar with the theorem, Saari gives a pretty comprehensive coverage of what it is, how it is derived and its implications.

A few key takeaways that I got from this book.
1 - In describing Arrow's theorem, he clearly demonstrates with examples how no voting system is universally the best. He provides a wide variety of simple examples to explain the paradoxes that result from Arrow's and Sens's famous theorems.
2 - He explains why the Borda Count is the best voting method, over plurality voting (what we typically experience), pairwise comparisons like Condorcet and others. He makes a good case for it but other scholars have critique the Borda approach as being easily manipulated by voters.
3 - He shows how these voting methods which only look at a subset of comparisons at a time are flawed and makes comparisons to other similar topics such as economics and engineering. He is a big fan of looking at the big picture.