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  • Voting power (7): Quarreling voters

    Jan 24, 2021 voting algebra

    In all the previous discussions of voting power, we have assumed that all winning coalitions are equally likely. But in practice that is not necessarily the case. Two or more voters may be opposed on so many issues that they would never vote the same way on any issues: such a pair of voters may be said to be …

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  • Voting power (6): Polynomial rings

    Jan 22, 2021 voting algebra

    As we have seen previously, it's possible to compute power indices by means of polynomial generating functions. We shall extend previous examples to include the Deegan-Packel index, in a way somewhat different to that of Alonso-Meijide et al (see previous post for reference). Again, suppose we consider the voting game …

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  • Voting power (5): The Deegan-Packel and Holler power indices

    Jan 14, 2021

    We have explored the Banzhaf and Shapley-Shubik power indices, which both consider the ways in which any voter can be pivotal, or critical, or necessary, to a winning coalition. A more recent power index, which takes a different approach, was defined by Deegan and Packel in 1976, and considers only minimal winning …

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  • Three-dimensional impossible CAD

    Jan 10, 2021 geometry CAD

    Recently I friend and I wrote a semi-serious paper called "The geometry of impossible objects" to be delivered at a mathematics technology conference. The reviewer was not hugely complimentary, saying that there was nothing new in the paper. Well, maybe not, but we had fun pulling together some information …

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  • Voting power (4): Speeding up the computation

    Jan 6, 2021 voting algebra julia

    Introduction and recapitulation Recall from previous posts that we have considered two power indices for computing the power of a voter in a weighted system; that is, the ability of a voter to influence the outcome of a vote. Such systems occur when the voting body is made up of a number of "blocs": these may …

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  • Voting power (3): The American swing states

    Jan 3, 2021 voting algebra

    As we all know, American Presidential elections are done with a two-stage process: first the public votes, and then the Electoral College votes. It is the Electoral College that actually votes for the President; but they vote (in their respective states) in accordance with the plurality determined by the public vote. …

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  • Voting power (2): computation

    Dec 31, 2020 voting algebra

    Naive implementation of Banzhaf power indices As we saw in the previous post, computation of the power indices can become unwieldy as the number of voters increases. However, we can very simply write a program to compute the Banzhaf power indices simply by looping over all subsets of the weights: def banzhaf1(q,w): n = …

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  • Voting power

    Dec 30, 2020 voting

    After the 2020 American Presidential election, with the usual post-election analyses and (in this case) vast numbers of lawsuits, I started looking at the Electoral College, and trying to work out how it worked in terms of power. Although power is often conflated simply with the number of votes, that's not necessarily …

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  • Electing a president

    Nov 7, 2020 voting linear-programming

    Every four years (barring death or some other catastrophe), the USA goes through the periodic madness of a presidential election. Wild behaviour, inaccuracies, mud-slinging from both sides have been central since George Washington's second term. And the entire business of voting is muddied by the Electoral College, the …

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  • Enumerating the rationals

    Jan 18, 2020 mathematics

    The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this: \[ \left[\begin{array}{ccccc} …

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Recent Posts

  • Voting power (7): Quarreling voters
  • Voting power (6): Polynomial rings
  • Voting power (5): The Deegan-Packel and Holler power indices
  • Three-dimensional impossible CAD
  • Voting power (4): Speeding up the computation
  • Voting power (3): The American swing states
  • Voting power (2): computation
  • Voting power
  • Electing a president
  • Enumerating the rationals

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