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 …

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 …

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 …

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 …

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. …

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: 1def banzhaf1(q,w): 2 …

Here we show how a Tschirnhausen transformation can be used to solve a quartic equation. The steps are: Ensure the quartic is missing the cubic term, and its initial coefficient is 1. We can do this by first dividing by the initial coefficient to obtain an equation \[ x^4+b_3x^3+b_2x^2+b_1x+b_0=0 \] and then replace …

A general cubic polynomial has the form \[ ax^3+bx^2+cx+d \] but a general cubic equation can have the form \[ x^3+ax^2+bx+c=0. \] We can always divide through by the coefficient of \(x^3\) (assuming it to be non-zero) to obtain a monic equation; that is, with leading coefficient of 1. We can now remove the \(x^2\) …

These are a class of root-finding methods; that is, for the numerical solution of a single nonlinear equation, developed by Alston Scott Householder in 1970. They may be considered a generalisation of the well known Newton-Raphson method (also known more simply as Newton's method) defined by \[ x\leftarrow …

When I was teaching the binomial theorem (or, to be more accurate, the binomial expansion) to my long-suffering students, one of them asked me if there was a trinomial theorem. Well, of course there is, although in fact expanding sums of greater than two terms is generally not classed as a theorem described by the …