SAVE Voters and Electorates

The answer to that question depends on the choice being made. For example, consider a community considering investing in its own cell phone service. In this case, each dimension is represented by the possible answers to various questions:

• What is the cost of the system?
• How will it be funded?
• What is the service area?
• What is the minimum quality of the service?
• What is the most appropriate technology?
• What impact will this have on our community and environs?

Each voter would consider these questions, and probably others, and have their own preferences depending on their own opinions and expertise. At some point, various possible actions would be proposed, which will be called motions. These motions will all be different along one or more of the dimensions. If there are no differences between two motions, the two are considered the same motion. Any differences between motions can cause them to be considered different, but differences only matter if those differences are considered important to some voter.

Some dimensions are discrete, such as the number of units necessary for proper coverage of the service area; it isn't possible to have one and a half cell towers. Other dimensions might involve a range of possibilities, such as costs. Sometimes questions span multiple dimensions, such as where to locate a cell tower, which is a two-dimensional question.

In addition to motions, there is also commentary. Any motion is about what to do. Actually implementing a motion will have consequences, some of which may be desired and intentional, while others may be unintended or undesirable.

All of this needs to be represented as part of the spatial model.

Yet the model needs to be tractable. Thus for the purposes of the model, a motion includes both the actual proposal (what is to be done), and its commentary (opinions, testimony, propaganda). Everything about a motion, what it is claimed to do by its proposers, what those proposers think it actually will do, what both critics and supporters fear and hope it will do, if indeed it will do anything at all. All of this is a multi-dimensional part of the motion. And the same is the case for each and every motion. No matter what the voting system is.

about Voters can be exposed to all sorts of perspectives on the motions, and it is up to the voter to

These are all objective and measurable. Voters have ideal preferences as to the final outcome, which include every dimension included in a motion (including commentary) and further include weights on each dimension.

Thus the spatial model will have \(N\) dimensions, and:

• A motion is represented as a single an N-dimensional vector:
  • motion: \(M = (m_{1},m_{2},\ldots,m_{N-1},m_{N})\).
• A voter is represented as a pair of N-dimensional vectors:
  • preference: \(P = (p_{1},p_{2},\ldots,p_{N-1},p_{N})\), and
  • weight: \(W = (w_{1},w_{2},\ldots,w_{N-1},w_{N})\), where
    • \(\forall i \in {1,2,\ldots,N}, w_{i} >= 0\), and
    • \(\Sigma_{i=1}^{i=N} w_{i} = N\).

Given the representations of motions and voters where both are points in an \(N\)-dimensional space, the ''natural'' thing to do is apply a metric function and consider \(\delta_{i} = |m_{i}-p_{i}|\) over all the \(i\) values. However, we also need to factor in the weights, so the revised values are: \(\delta_{i} = w_{i}|m_{i}-p_{i}|\) over all the \(i\).

This gives us a vector of weighted differences, but we need a bit more. In order to something like simulate a voter ranking a set of motions, we need to get a single, scalar value. With the data we have, there are an infinite number of metrics we could use, but there are six metrics that seem most useful. These are the weighted and unweighted variants of the first two Minkowski metrics and the limiting case of the Minkowski series, which is the Chebyshev metric.

In the Minkowski series of metrics, the partial differences are mapped by an exponent, summed, then reduced by the same exponent. For example, when the exponent is \(1\) the deltas are raised to the first power (i.e. left alone), summed, and raised to the \(-1\) power. When the exponent is \(2\), we have the familiar Euclidean metric. When the exponent is \(n\), the partials are raised to the \(n^{th}\) power, summed, and the \(n^th\) root of the sum is the distance. The effect of increasing the power is to place increasing weight to the largest partial, and at the limit, the Chebyshev metric, the only thing that counts is the largest partial difference.

The formulae for these six metrics, where \( I = {1,2,\ldots,N} \), are:

Unweighted Minkowski metric 1

\begin{equation} L_{1}(P,M) = \Sigma_{i\in{I}} |m_{i}-p_{i}| \end{equation}

Unweighted Minkowski metric 2

\begin{equation} L_{2}(P,M) = \sqrt{\Sigma_{i\in{I}} (m_{i}-p_{i})^2} \end{equation}

Unweighted Chebyshev metric

\begin{equation} L_{\infty}(P,M) = \max_{i\in{I}} (m_{i}-p_{i}) \end{equation}

Weighted Minkowski metric 1

\begin{equation} L_{1}(P,M) = \Sigma_{i\in{I}} w_{i}|m_{i}-p_{i}| \end{equation}

Weighted Minkowski metric 2

\begin{equation} L_{2}(P,M) = \sqrt{\Sigma_{i\in{I}} (w_{i}(m_{i}-p_{i}))^2} \end{equation}

Weighted Chebyshev metric

\begin{equation} L_{\infty}(P,M) = \max_{i\in{I}} w_{i}(m_{i}-p_{i}) \end{equation}

These six metrics can be describes as follows:

Unweighted Minkowski metric 1

This is also called the city block, taxicab or Manhattan metric. Distances are measured along a grid. A circle in this metric shows as a square with the sides of slope 1 or -1.

Unweighted Minkowski metric 2

This is the familiar Euclidean metric. A circle in this metric is the familiar circle from geometry.

Unweighted Chebyshev metric

This is an unusual but simple metric in which the only thing that counts is the largest partial difference. A circle in this metric looks like a square with sides that are either horizontal or vertical.

Including the weights distorts the distances by compressing the partials with more weight and stretching the partials with less weight. If the ratio of the weight given to the X and Y dimensions is \(2:3\), the weight of the X value is \(0.8\), and the corresponding Y weight is \(1.2\).

Weighted Minkowski metric 1

In this weighted variant of \(L_{1}\), the weights given above would show a unit circle as a rhombus with corners at \((-0.8, 0)\), \((0, 1.2)\), \((0.8, 0)\), and \((0, -1.20)\).

Weighted Minkowski metric 2

In this weighted variant of the familiar Euclidean metric, the above weights would show a unit circle as an ellipse centered at the origin, with a horizontal minor axis of length \(1.6\), and a vertical major axis of length \(2.4\).

Weighted Chebyshev metric

In this weighted variant of the Chebyshev metric, the weights above would show a unit circle as a rectangle, centered at the origin, with a width of \(1.6\) and a height of \(2.4\), with sides that are parallel to the axes.

The initial image using the \(L_{2}\) metric with weights, zero tolerance, and an X weight multiplier in the \([0.01-1.99]\) range, shows a voter preference position (displayed as a yellow oval with a black border), three motions (displayed as smaller yellow circles with a black border) labeled with the letters A, B, and C, and an oval curve centered on the voter position with the same axis ratios as the voter and running through one of the motions such that of the remaining two motions, one is inside the oval and the other is outside the oval.

Describe the picture: labeled motions, a voter, and an indifference curve. The voter shape and the indifference curve are determined by the active metric.

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