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.
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Notes
This holds only for n = 3. For n ≥ 4, there are 2n orthants in Rn. The geometric assignment of n! orthants for the Condorcet winners leaves behind a region so divided that it is impossible to combine them into the required convex sets. (The problem is worse when the regions of coordinate planes and axes are introduced [S12].) Thus, constructing a weakly consistent extension of the Condorcet winner based on rankings, is doomed for n ≥ 4.
For the reader familiar with calculus, this means that the first two derivatives of G are defined and continuous. For the reader still waiting to be initiated into the Calculus Club, treat G as a function where the graph can be smoothly drawn.
Of course, Black was unaware of the confused voter bias of the Condorcet winner; a bias that casts serious doubt on his procedure.
A tie breaking scheme depending on vote totals rather than ranking requires us to examine the set of supporting profiles.
For the resider who knows the terms, observe that specifying a line emanating from a point and analyzing the changes corresponds to the “directional derivative” of the procedure.
This “middle-ranked involvement” includes the “abstention” paradox which seems to have been discovered by Smith [Sm], significantly advanced for the plurality runoff by Brains and Fishburn [BF3], and then described in more general settings by Saari [S9].
[He] It is clear from the publication date of this book (1940) that this warning was written before the true consequences of these comments were known.
In practice, the redistribution of votes is done in many ways. A silly way is to stop counting votes for a candidate as soon as she wins; on all remaining ballots her name is ignored and all candidates move up one position. Cincinnati used a clever approach; they used a formula to determine how many ballots needs to be redistributed. Then, they counted through the ballots in multiples of this number to adjust certain rankings. For details, see [].
The political party usually determines the ordering of the candidates on a list. Alternatively, such as in Brazil, the voters can influence this ordering by voting for particular candidates.
In part, this is because with the population of the day Hamilton’s method would award Connecticut more representatives than permitted by the Constitutional bound of no more than one representative per 30,000 persons. See [Mon, Mas, BY1].
For the mathematicians, this could be taken in the sense of Lebesgue measure, or as being an open-dense set. Actually, by examining the proof, more exacting algebraic requirements of excluding a finite number of points can be imposed.
Indeed, the set of p ∈ Si(m) that generates an Alabama Paradox includes an open dense set.
Other issues, such as when subcommittees unite, are left to the reader. Such questions have relevance for those countries using the “list” PR method. For instance, if the outcome in two regions places party 1 in the top-ranks, must the combined outcome also favor this party?
No serious problems occur for m = 2, but pictures are easy to draw.
Regional voting in Congress is becoming increasingly important for many policy factors. Indeed, for many issues, the concerns of a region overtake those of the individual states. Wendy Ramsbottom explored this question (in her 1993 Northwestern University Senior Thesis) by comparing the number of representatives from all states in a region with the number of representatives the region would receive if it were a state. In this way, she demonstrated that certain regions are short-changed in political power; a factor that can be critical on a close vote. (She identified issues where this could have been a deciding factor in several Congressional votes.) The natural question raised by her work is whether a method of the above kind can keep both kinds of apportionments — state and region — as consistent as possible. This is the Borda issue.
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© 1995 Springer-Verlag Berlin Heidelberg
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Saari, D.G. (1995). Other Voting Issues. In: Basic Geometry of Voting. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57748-2_5
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DOI: https://doi.org/10.1007/978-3-642-57748-2_5
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