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## Über dieses Buch

Over two centuries of theory and practical experience have taught us that election and decision procedures do not behave as expected. Instead, we now know that when different tallying methods are applied to the same ballots, radically different outcomes can emerge, that most procedures can select the candidate, the voters view as being inferior, and that some commonly used methods have the disturbing anomaly that a winning candidate can lose after receiving added support. A geometric theory is developed to remove much of the mystery of three-candidate voting procedures. In this manner, the spectrum of election outcomes from all positional methods can be compared, new flaws with widely accepted concepts (such as the "Condorcet winner") are identified, and extensions to standard results (e.g. Black's single-peakedness) are obtained. Many of these results are based on the "profile coordinates" introduced here, which makes it possible to "see" the set of all possible voters' preferences leading to specified election outcomes. Thus, it now is possible to visually compare the likelihood of various conclusions. Also, geometry is applied to apportionment methods to uncover new explanations why such methods can create troubling problems.

## Inhaltsverzeichnis

### Chapter I. From an Election Fable to Election Procedures

Abstract
What could be easier than interpreting an election? Even nursery school children use voting to select their juice of choice before nap time. After all, it just involves counting where the candidate with the most votes wins. What is so difficult about something as elementary as that?
Donald G. Saari

### Chapter II. Geometry for Positional and Pairwise Voting

Abstract
A plurality election is simple; just count how many voters have each candidate top-ranked. It is surprising that using this elementary description to analyze the procedure quickly introduces mathematical complications that severely limit what can be learned.
Donald G. Saari

### Chapter III. From Symmetry to the Borda Count and Other Procedures

Abstract
In this chapter I explore how geometric symmetry helps explain voting outcomes and procedures. For instance, the beverage example proves that the pairwise votes can reverse the plurality outcome, but why? The Condorcet profile shows that pairwise votes can create a cycle, but why? Certain profiles, such as a division between voters of types-one and five, create problems with voting procedures, but why? Some w s procedures seem better than others, but is this true? In the first technical section, I show why some procedures lack certain symmetry properties; the cost is demonstrated with new paradoxes. The remaining sections return to the analysis of particular procedures.
Donald G. Saari

### Chapter IV. Many Profiles; Many New Paradoxes

Abstract
I now turn from the single profile consequences of election outcomes to describe the fascinating properties of voting theory involving several profiles. A natural example is the electoral fable controversy about the Dean’s Council caused by combining the two profiles — one for each subcommittee. Beyond constructing amusing “paradoxes,” the importance of multiprofile issues is, for instance, to understand what can happen if a voter votes strategically, or if he doesn’t vote. (The voter’s options are to vote sincerely, strategically, or abstain; each option defines a different profile.) Other multiprofile issues include a concern about the consequences should more voters vote. What happens if voters change preferences? Can forming a coalition cause problems? In fact, as I show, important theorems in social choice theory, such as the Arrow Impossibility Theorem, are based on the properties a procedure must exhibit with changes in profiles.
Donald G. Saari

### Backmatter

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