Voter Metrics

Serial Approval Vote Election (SAVE)

A new voting system for finding consensus

Explore Voter metrics

Voters Metrics - Examples of Possible Metrics ,,fold,,

These spatial models represent voters and motions in the same "space" and show why any voter would have a particular preference order for the given motions. Voters prefer motions closer to their individual ideals (their position in the space) to motions further away.

A metric is another word for a distance function, and their are three general metrics used in these models. These are different ways of measuring the distance between two points. For example, given points \(A\) and \(B\), at coordinates \((1,2)\) and \((4,6)\) respectively:

The taxicab¹ metric takes the difference in the X values (\(4-1=3\)), and the difference in the Y values (\(6-2=4\)) and adds them together to get \(7\). This is the length of shortest path between \(A\) and \(B\) when the path segments are limited to horizontal and vertical line segments.
The Euclidean² metric is the length of a straight line segment connecting the two points. It looks at the X difference of \(3\) and the Y difference \(4\), and treats them as the sides of a right triangle, and uses \(\sqrt{3^2+4^2}=5\) to get \(5\), the hypotenuse of the triangle.
The Chebyshev³ metric is the longest of the coordinate distances between two points. For our \(A\) and \(B\) example, the X distance is \(3\), and the Y distance is \(4\), so the smaller X distance is completely ignored and the Chebyshev distance is \(4\).

For these three general metrics, all voters agree on the distances. However, in a voting context the X and Y coordinates are not necessarily related. If your issues are budgetary, such as how much to spend on defense and education, and motions are ordered pairs of the two spending amounts, the taxicab metric is the best metric to use in this case because the two parts of the motion are independent. It makes no sense to talk about the hypotenuse of defense and education, so the Euclidean metric is not appropriate, and the Chebyshev metric simply ignores the smaller change, so probably also is not the right general metric.

The other three metrics use relative weight values that are specific to each voter and the distances they calculate are thus subjective. The weights are applied to the individual issue differences before they are combined to the final distance measure. Objectively, motions \(A\) and \(B\) have exact values, common to all voters. But a voter considering a motion voter is not concerned with distance between motions, but rather the distance between a motions and the voter's ideal; something specific to the voter. Moreover, with a weighted metric, each coordinate difference has a different weight. A parent with school-age children might consider education spending to be more important than defense spending, while that same parent a few years later when there children are old enough to join the military, may decide changes in defense spending are more important.

The table below shows the symbol⁴ and formula for each of the six available metrics.

metric name	symbol	distance formula
taxicab	\(L^1\)	\(d = \lvert\Delta{x}\rvert + \lvert\Delta{y}\rvert\)
weighted taxicab	\(L^{1}_w\)	\(d = \lvert\Delta{x}w_x\rvert + \lvert\Delta{y}w_y\rvert\)
Euclidean	\(L^2\)	\(d = \sqrt{\Delta{x}^2+\Delta{y}^2}\)
weighted Euclidean	\(L^{2}_w\)	\(d = \sqrt{(\Delta{x w_x})^2+(\Delta{y w_y})^2}\)
Chebyshev	\(L^{∞}\)	\(d = \max(\lvert\Delta{x}\rvert, \lvert\Delta{y}\rvert)\)
weighted Chebyshev	\(L^{∞}_w\)	\(d = \max(\lvert\Delta{x}w_x\rvert, \lvert\Delta{y}w_y\rvert)\)

All of the weights \(w_x\),\(w_y\) in the above table are constrained such that \(w_x\) is a positive integer multiple of \(0.01\), \(0 < w_x <2\), and \(w_y = 2 - w_x\).

While these equations are exact, this explorable is presented to familiarize you with how voters are represented in later SAVE explorables. The main explanatory explorables only use \(2\) dimensions, and how (or whether) the issue differences combine for each voter will depend on the metric.

Description of Simulation Controls ,,fold,,

This explorable has a single voter and \(3\) motions. All of the items may be selected and moved about. Initially and as you change things, the explorable updates according to the following rules:

After any change, the voter recalculates the distances to the three motions.
The distances are sorted, and an indifference curve based on the active metric is drawn centered on the voter and passing through the middle distance motion.
The tolerance factor is applied to the curve as a shaded region that thickens the curve.
If any motions other than the middle distance motion is inside the tolerance band, that motion is marked as similar or equivalent to the middle motion as far as the preference order is concerned.
The caption is updated to show the new active metric and adjust the preference order.

The voter metric controls let you:

Use the Metric radio buttons to select one of:
- the taxicab or \(L^{1}\) metric,
- the normal Euclidean or \(L^{2}\) metric, or
- the Chebyshev or \(L^{\infty}\) metric.
Toggle the use of weights using the Use weights checkbox.
Manually adjust the tolerance percentage.
Manually adjust voter weights for the X and Y partial differences.
Toggle the restriction of motions to be on a line.

The position controls have been expanded. Selection (and deselection) is still done via clicks in the display area. Once the voter or any motions is selected:

Drag the handle to reposition the selected item, or
Edit the X and Y position readouts to set the new position.

Voter metric explorable

Figure 1: A voter with three motions to choose from, under various metrics.

Things to look for in this simulation ,,fold,,

The main things to get from this explorable is how changes in the active metric or the weights of a voter can change the preference order of the motions.

If you want a bit of a challenge, note that simple trial and error positioning and a particular weight setting can get you to a setup where \(5\) of the \(6\) possible strict motion preference orders can be seen just by running through the six metric settings.

Is it possible to position the motions and voter, and set up the weights so you can get all six strict preference orders just by changing the metrics?

I do not know the answer to this question, but my current hypothesis is that it is not possible. I'd be delighted if you could come up with a counter-example.

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 ellipse 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 ellipse centered on the voter position with the same axis ratios as the voter and running through the center of one of the motions such that of the remaining two motions, one is inside the ellipse and the other is outside the ellipse.

The element centered on the voter and intersecting the middle distance motion is called an indifference curve, and indicates the positions that are the same distance from the voter as the middle motion under the active metric. The points on the indifference curve are equivalent to the voter in that no point on the curve is considered better than any other point. The indifference curve divides the region into \(3\) parts: the curve itself, the points outside the curve that are all subjectively worse than any point on the curve, and the points inside the curve that are all subjectively better than any points on or outside the curve.

Footnotes:

To learn more about the taxicab metric visit the Wikipedia page on Taxicab geometry.

To learn more about the Euclidean metric, visit the Wikipedia page on Euclidean distance.

To learn more about the Chebyshev metric, visit the Wikipedia page on Chebyshev distance.

⁴

The symbol is from the Minkowski distance classification. To learn more see the Wikipedia page Minkowski distance.