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Rating: 
Summary: The most important work since Arrow
Review: This book is the most important work in social choice theory since Arrow's (1963) "Social Choice and Individual Values". Professor Saari (now at UC Irvine) used this book in an advanced graduate course I took in Fall 2000, and he covered nearly the entire book in a ten week course (hint to instructors and students: I would not recommend this suicidal pace, unless your students are very ambitious and/or very bright!) The goal of the book is ambititous, and yet very simple. One of the biggest difficulties with voting theory and social choice is the "curse of integers or discreteness" - when we consider more than three alternatives, the number of alternative arrangements of voter preferences escalates quickly. This means that the main ideas in voting theory cannot usually be represented or analyzed by drawing a picture or using calculus, unlike most ideas in economics (eg the Edgeworth Box, demand/supply etc). Saari avoids this problem by working with continuous spaces; he uses the geometry of the unit simplex (a familiar tool for most economics grad students) and the unit cube to analyze and explain just about all of the most important issues and results in social choice theory: cycling, manipulation, voting paradoxes, Arrow's theorem, Sen's theorem, the Gibbard-Satterthwaite theorem, and much, much more. But the geometric approach is not just a cute pedagogic tool. On the contrary, the methods in this book allow researchers to state and prove new conjectures about voting methods using standard ideas from calculus, linear algebra, and basic high-school geometry; without these tools new results would be nearly impossible to even state, let alone prove. The writing style is mostly informal, and many statements are not proved rigorously (they have only just recently appeared in the professional journal literature). Depending on your background this is either good or bad; but those with a graduate math background (like myself) can just go to the journals to find the proofs of statements if they so desire. Probably the best part of the book is that there is a massive collection of problems at the end of each section - and many of the these problems are research questions in their own right. The other fun part of the book is that once you have learned to use Saari's geometric tools, you can create just about any crazy voting paradox you like in a couple of minutes, whereas this previously might have taken months or would constitute an entire research project. Therefore the book is very stimulating for advanced grad students and researchers in the field, as well as those encountering social choice theory for the first time. Grad students - as a final inducement for reading this book, once you learn Saari's tools, you will be able to embarass about 99% of your professors and fellow students with your newly acquired skills. It is easy to find questions that are simply impossible to answer without using Saari's tools. I would recommend the book to advanced graduate students in economics, mathematics and political science, and researchers in those fields.
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