Basic Geometry of Voting [Paperback]

Donald G. Saari (Author)

Book Description

Series: Basic Geometry of Voting | Publication Date: January 29, 2003

Amazingly, the complexities of voting theory can be explained and resolved with comfortable geometry. A geometry which unifies such seemingly disparate topics as manipulation, monotonicity, and even the apportionment issues of the US Supreme Court. Although directed mainly toward students and others wishing to learn about voting, experts will discover here many previously unpublished results. As an example, a new profile decomposition quickly resolves the age-old controversies of Condorcet and Borda, demonstrates that the rankings of pairwise and other methods differ because they rely on different information, casts serious doubt on the reliability of a Condorcet winner as a standard for the field, makes the famous Arrow's Theorem predictable, and simplifies the construction of examples.

Editorial Reviews

Review

My review of Geometry of Voting was enthusiastic. A last comment: If you have a copy of it, get a copy of Basic Geometry of Voting also; if you haven't, get both. -- Maurice Salles, Mathematical Reviews, issue 98d

Product Details

• Paperback: 312 pages
• Publisher: Springer; 1 edition (January 29, 2003)
• Language: English
• ISBN-10: 3540600647
• ISBN-13: 978-3540600640
• Product Dimensions: 9.1 x 5.9 x 0.8 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,236,459 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

17 of 17 people found the following review helpful:

5.0 out of 5 stars The most important work since Arrow, January 18, 2001

By A Customer

This review is from: Basic Geometry of Voting (Paperback)

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.

2 of 2 people found the following review helpful:

5.0 out of 5 stars learn mathematics of voting, September 10, 2005

By 

M. Holmes - See all my reviews
(REAL NAME)   

This review is from: Basic Geometry of Voting (Paperback)

Great book! Interesting new theory developed to visualize voting systems. Can be technical - best used in conjunction with "Chaotic Elections", which is more of an overview. It is nice to see an application of mathematics that doesn't require a huge amount of mathematical training - just some familiarity with vectors and parametrizations of lines and planes. Accessible to many people interested in how math can be used to model voting systems, from high school onwards.