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Basic Geometry of Voting [Paperback]

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Item description for Basic Geometry of Voting by Donald G. Saari...

A surprise is how the complexities of voting theory can be explained and resolved with the comfortable geometry of our three-dimensional world. This book is directed toward students and others wishing to learn about voting, experts will discover previously unpublished results. As an example, a new profile decomposition quickly resolves two centuries old controversies of Condorcet and Borda, demonstrates, that the rankings of pairwise and other methods differ because they rely on different information, casts series 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. The geometry unifies seemingly disparate topics as manipulation, monotonicity, and even the apportionment issues of the US Supreme Court.

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Item Specifications...

Pages   300
Est. Packaging Dimensions:   Length: 1" Width: 6.25" Height: 9.25"
Weight:   1.1 lbs.
Binding  Softcover
Release Date   Jan 29, 2003
Publisher   Springer
ISBN  3540600647  
ISBN13  9783540600640  

Availability  0 units.

More About Donald G. Saari

Register your artisan biography and upload your photo! Donald G. Saari is Distinguished Professor of Mathematics and Economics and Honorary Professor of Logic and Philosophy of Science at the University of California Irvine, where he is Director of the Institute for Mathematical Behavioral Sciences. He previously served on the faculty of Northwestern University from 1968 to 2000, where he held the Pancoe Professorship of Mathematics. A Member of the U.S. National Academy of Sciences, Fellow of the American Academy of Arts and Sciences, and Fellow of the American Association for the Advancement of Science, Professor Saari is the former Chief Editor of the Bulletin of the American Mathematical Society. The author of more than 170 published papers, he has also written numerous books, including Basic Geometry of Voting (1995), Decisions and Elections: Explaining the Unexpected (Cambridge University Press, 2001), Chaotic Elections! A Mathematician Looks at Voting (2001), The Way It Was: Mathematics from the Early Years of the Bulletin (2003), and Collisions, Rings, and Other Newtonian N-Body Problems (2005).

Donald G. Saari has an academic affiliation as follows - University of California, Irvine.

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Reviews - What do customers think about Basic Geometry of Voting?

learn mathematics of voting  Sep 11, 2005
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.
The most important work since Arrow  Jan 19, 2001
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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