Mathematics and Democracy: Recent Advances in Voting Systems by Fuad Aleskerov (auth.), Prof. Dr. Bruno Simeone, Prof. Dr.

By Fuad Aleskerov (auth.), Prof. Dr. Bruno Simeone, Prof. Dr. Friedrich Pukelsheim (eds.)

From the reviews:

"The … booklet is an edited quantity, and indicates either the issues and virtues of that style. … i discovered the ebook attention-grabbing and demanding. … the ebook is definitely worthy having in your shelf. … it might be necessary, actually, as a lecture room workout for college students in various disciplines, since it either illustrates the unnatural energy of arithmetic to light up difficult questions and unites a number of it appears unrelated difficulties in one analytic rubric. … it is a worthy and demanding book." (Michael Munger, Public selection, Vol. 132, 2007)

"This is an exceptional selection of 17 papers selected from contributions to the foreign Workshop on arithmetic and Democracy: balloting structures and Collective selection that happened in Erice, Sicily in 2005. … one will stroll clear of this quantity with the experience that one has been uncovered to the learn frontier of the electoral approach theory." (Ugur Ozdemir, Social selection and Welfare, Vol. 31, 2008)

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By Fuad Aleskerov (auth.), Prof. Dr. Bruno Simeone, Prof. Dr. Friedrich Pukelsheim (eds.)

From the reviews:

"The … booklet is an edited quantity, and indicates either the issues and virtues of that style. … i discovered the ebook attention-grabbing and demanding. … the ebook is definitely worthy having in your shelf. … it might be necessary, actually, as a lecture room workout for college students in various disciplines, since it either illustrates the unnatural energy of arithmetic to light up difficult questions and unites a number of it appears unrelated difficulties in one analytic rubric. … it is a worthy and demanding book." (Michael Munger, Public selection, Vol. 132, 2007)

"This is an exceptional selection of 17 papers selected from contributions to the foreign Workshop on arithmetic and Democracy: balloting structures and Collective selection that happened in Erice, Sicily in 2005. … one will stroll clear of this quantity with the experience that one has been uncovered to the learn frontier of the electoral approach theory." (Ugur Ozdemir, Social selection and Welfare, Vol. 31, 2008)

Show description

Read or Download Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice PDF

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Additional resources for Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice

Sample text

In addition, if a change could be made that agrees with Φ2 then it would yield another apportionment of Φ. Theorem 1 The unique method Φ for vector problems that is coherent with Φ2d is the divisor method Φd . Proof. The condition is necessary, since Φd is obviously coherent with Φ2d . To see that the condition is sufficient, suppose Φ is any method that is coherent with Φ2d and that a ∈ Φ(v, h). It is shown that a ∈ Φd (v, h). Choose λ > 0 such that i [λvi ]d = h, and let bi = [λvi ]d , so b ∈ Φd (v, h), implying, in particular, that (bi , bj ) ∈ Φ2d (vi , vj ), bi + bj for every pair i, j.

Moreover we say that a district design π is (blue) extremal if the number of blue districts b(ω, π) is equal to its upper bound n/(s + 1) . Similar concepts can be introduced for the red party. Remark 3 If p ≤ s + 1, each blue extremal partition has p − 1 blue districts and one red district. We are especially interested in the following optimization problem: GAP (G) = max(max b(ω, π) − min b(ω, π)). ω∈Ω π∈Π π∈Π For a given graph G the function GAP (G) is a measure of the maximum bias of an electoral outcome in terms of number of seats in single member majority districts.

Since a party i (or row) deserves a fixed number of seats ri , rescaling by multiplying its votes by λi > 0 should (as in the vector problem) change nothing; symmetrically, since a region j is assigned a fixed number of seats cj , rescaling its votes by µj > 0 should (as in the vector problem) change nothing as well. Property 4 A method Φ for matrix problems is said to be proportional if Φ(v, r, c) = Φ(λ ◦ v ◦ µ, r, c) for every real λ, µ > 0. Given any two apportionments a, b of a problem (v, r, c), consider a − b.

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