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Über dieses Buch

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 This book is directed toward students and others wishing to learn about voting, experts will discover previously unpublished results.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.

Inhaltsverzeichnis

Frontmatter

Chapter I. From an Election Fable to Election Procedures

Abstract
What could be easier than interpreting an election? You just count to determine which candidate receives the most votes. What can be difficult about something so elementary? After all, even nursery school children know how to vote to select their juice of choice before nap time.
Donald G. Saari

Chapter II. Geometry for Positional and Pairwise Voting

Abstract
It is trivial to tally a plurality election; just count how many voters have each candidate top-ranked. Surprisingly, when this elementary description is used to analyze the procedure, it quickly introduces mathematical complications that severely limit what can be learned.
Donald G. Saari

Chapter III. The Problem with Condorcet

Abstract
The beverage brouhaha, which initiated the season of dissent for the hypothetical department of the fable, started with the winer’s discovery that the department’s plurality ranking conflicts with their rankings of pairs of beverages. The radical disagreement raises interesting theoretical questions. How does a majority vote ranking of a pair relate to its relative ranking within a plurality outcome? Can anything go wrong with pairwise rankings?
Donald G. Saari

Chapter IV. Positional Voting and the BC

Abstract
As we learned in Chap. 3, problems and paradoxes must be anticipated with procedures based on pairwise rankings. The subtle reason is that these methods manage to drop the central assumption of transitivity. This suggests using methods, such as the plurality vote and the BC, which admit only rational voters. Do they have problems? Rest assured; there are plenty of them! But, with a further decomposition of profiles, it becomes easy to analyze the problems and create profile examples.
Donald G. Saari

Chapter V. Other Voting Issues

Abstract
In addition to the single profile consequences, there are fascinating voting properties which require several profiles. A natural illustration is the Dean’s Council controversy (in the fable) caused by combining the profiles for two subcommittees. Multiprofile issues are important because they help us understand what can happen if a voter votes strategically, or if he doesn’t vote, or whether there are problems should new voters arrive, or when groups combine to form coalitions, or if voters change preferences, or … In Sect. 5.1, problems such as the Dean’s council are explained. In Sect. 5.2, attention turns to how “more can be less.” In Sect. 5.3, the emphasis is on strategic voting. While all of these conclusions depend upon the geometry of profile sets, whenever possible, the simpler geometry of the representation triangle or cube is exploited. Then, to conclude, Sects. 5.4, 5.5 address the intriguing problems of the “list” methods and apportionment of Congressional seats.
Donald G. Saari

Notes

Abstract
By no means are the following references complete; they are places to start. In addition to these suggestions and their references, I recommend Kelly’s Social Choice Bibliography [K3].
Donald G. Saari

References

Without Abstract
Donald G. Saari

Backmatter

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