Book-Basic Geometry of Voting
DEMOREP1 at aol.com
DEMOREP1 at aol.com
Tue May 28 20:59:15 PDT 1996
The below appears as one item if +voting+methods is used in a search using
the altavista search engine at---
http://www.altavista.digital.com/cgi-bin/query?urlencoded_query_string
I have not seen the book. The below comments are the publishers or Mr.
Saari's. The book page is at----
http://math.nwu.edu/~d_saari/geom/geometry.html
------
Basic Geometry of Voting
Donald G. Saari
ISBN3-540-60064-7
Springer-Verlag (December, 1995)
Click to see a bigger cover
While this book has large number of new results about voting theory, the
emphasis purposely is placed on three candidate settings so that the book can
be read by almost everyone. (With four or more candidates, the results need
more advanced mathematical techniques.) Each section is followed by exercises
to facilitate the use of the book in the classroom.
Chapter 1. From an election fable to election procedures
Problems of election procedures are introduced via a story that weaves
together a series of unexpected election paradoxes. Then, some of the history
of voting is related (along with problems encountered by the Catholic Church
while trying to elect a Pope) to further describe the issues and motivate the
need to address these concerns.
Chapter 2. Geometry for positional and pairwise voting
It is shown how simple geometry of three-dimensional space can be used to
capture properties of individual preferences and election rankings.
Chapter 3. The problem with Condorcet
The Condorcet winner is the candidate who beats all others in a
"head-to-head" pairwise comparison. There is no question that this procedure,
with its sense of rugged individualism, is appealing. Indeed, the Condorcet
winner is widely used as a standard for the field of voting theory to measure
the merits of other methods. In this chapter, simple geometry is used to
explain why the Condorcet winner actually is a lousy concept! Then it is
shown how the source of the problems with Condorcet approach end up
explaining voting cycles. (The is where A beats B, and B beats C. While this
seems to imply that A should beat C, we have, instead, that C beats A.) It is
shown that cycles merely reflect the fact that "pairwise voting" kills the
assumption that voters have transitive preferences.
These arguments are then used to resolve the Borda-Condorcet debates that
started in the 1780s and continue today. This is done by analyzing the
profiles Condorcet created in his attempt to discredit Borda; it turns out
that just the opposite is true - the analysis of these examples show why
Condorcet's approach is inferior and his conclusions were wrong. Finally, it
is shown how this observation extends to cast doubt on the importance of
Arrow's well known theorem.
Chapter 4. Positional voting and the BC
This chapter introduces a new (geometric) approach to analyze positional
voting. For instance, the plurality vote is where one point is assigned to a
voter's top-ranked candidate and zero to all others. The Borda Count (BC) is
where two points are assigned to a top-ranked candidate, one to the
middle-ranked candidate, and none to the bottom-ranked candidate. There are,
of course, an infinite number of ways to assign points. The consequences of
doing this are explained. For instance, it is shown that the way in which
ballots are tallied can critically determine the election outcome; e.g.,
there exist simple examples where, by changing these weights, up to seven
different elections outcomes result.
Chapter 5. Other voting issues
With geometry, it becomes particularly easy to understand the many perplexing
problems of this area such as where after a winning candidate receives added
support, she loses, or the important Gibbard-Satterthwaite Theorem specifying
that all voting methods can be manipulated. The last two sections of the
chapter introduce a new, geometric approach to understand the serious
difficulties created by apportionment procedures of the type used to
determine the number of congressional seats to each state (based on the
census figures) and the proportional methods often used in Europe.
Related information and preprints can be found on Don Saari's home page.
More information about the Election-Methods
mailing list