Download e-book for iPad: Basic Geometry of Voting by Donald G. Saari

By Donald G. Saari

ISBN-10: 3540600647

ISBN-13: 9783540600640

ISBN-10: 3642577482

ISBN-13: 9783642577482

A shock is how the complexities of vote casting thought may be defined and resolved with the comfy geometry of our three-d global. This booklet is directed towards scholars and others wishing to profit approximately vote casting, specialists will realize formerly unpublished effects. for instance, a brand new profile decomposition fast resolves centuries outdated controversies of Condorcet and Borda, demonstrates, that the ratings of pairwise and different tools vary simply because they depend upon various info, casts sequence doubt at the reliability of a Condorcet winner as a regular for the sphere, makes the recognized Arrow`s Theorem predictable, and simplifies the development of examples. The geometry unifies likely disparate subject matters as manipulation, monotonicity, or even the apportionment problems with the U.S. superb Court.

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With his important contributions to weights, the pendulum, navigation, geodesy, scientific instruments, fluid flow - both theoretically and as applied to ships, serving on the commission to establish the metric system, his work earning his reputation as one of the founders of modern French mathematical physics, and on and on, it is easy to conjure an image of an introverted, absent-minded, sedentary academic balancing his time between a quiet study surrounded by books and a lab where he shuffled around attired in a stained lab coat.

The unanimity profile E i , where all voters are of the ith type, i = 1, ... , plays an important role. ) Just as the normalized outcome ej serves as a vertex for Si(3), the six unanimity profiles, {E;}~=l' are the vertices for Si(6). That is, Si(6) is the convex hull defined by these vertices (see Sect. 4). This fact is underscored by the obvious relationship 6 6 P = (PbP2,'" ,P6) = LpjEj , where LPj = l. 2) j=l Consequently, the Pi values are the convex weights defining the profile p. A similar statement holds for any number of candidates.

When an opportunity for a debate exists, it usually generates one - particularly in academic circles. " "Why not use, say, 4, 1, 0, or 5, 4, 17" Justifications for the BC weights were offered by Borda and by Laplace, but their philosophical responses are not satisfying; the questions remain. Another fault of Borda's method, as shown in the fable, is that the BC can be manipulated. The reported reaction from Borda asserting that his method is intended only for honest people is lame (Chap. 2 of [BI]); anyway, we now know that all methods can be manipulated.

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Basic Geometry of Voting by Donald G. Saari


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